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_pstyle54" -1 281 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle55" -1 282 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle29" -1 233 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle56" -1 283 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle30" -1 234 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle57" -1 284 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle5 8" -1 285 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }1 0 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle59" -1 286 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }1 0 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle60" -1 287 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle61" -1 288 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle 62" -1 289 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle31" -1 235 "Cour ier" 0 1 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle63" -1 290 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle64" -1 291 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle65" -1 292 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle32 " -1 236 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle66" -1 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 266 "" 0 "" {TEXT 210 62 "Enumerating unrooted ma ps using tree decomposition (Eric Fusy)" }{TEXT 210 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 333 "This worksheet illustrates the applic ation of the method of tree-decomposition to enumerate unrooted 2-conn ected and 3-connected maps on the sphere up to orientation-preserving \+ homeomorphism. The method is thoroughly exposed in the article \"enume rating unrooted maps using tree-decomposition\" available on the web p age of the author." }{TEXT 209 0 "" }}}{SECT 1 {PARA 268 "" 0 "" {TEXT 208 38 "Recall of the treatment of the article" }{TEXT 208 0 "" }}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 26 "Burnside-Liskovets Formula" }{TEXT 207 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 305 "Using Burnsi de lemma (adapted by Liskovets for the case of maps on the sphere), th e enumeration of a family of unrooted maps (for example the family of \+ maps without constraints but also the family of 2-connected or 3-conn ected maps) comes down to the enumeration of so-called k-rooted maps o f the family." }{TEXT 206 0 "" }}}}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 17 "Tutte's bijection" }{TEXT 207 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 689 "A well-known bijection associates a bicolored quadrangu lation to a map of the sphere. This bijection adapts well on 2-connect ed and 3-connected maps. More precisely, Tutte's bijection sets 2-con nected maps in bijection with quadrangulations without multiple edges, also called simple quadrangulations. It also sets 3-connected maps i n bijection with quadrangulation such that each 4-cycle delimits a fac e, these quadrangulations are said irreducible. Tutte's bijection adap ts to k-rooted objects. For example, it sets k-rooted 2-connected maps in bijection with k-rooted simple quadrangulations and sets k-rooted 3-connected maps in bijection with k-rooted irreducible quadrangulati ons." }{TEXT 206 0 "" }{TEXT 206 240 "\nAs a consequence the enumerati on of k-rooted maps, 2-connected maps, and 3-connected maps comes down respectively to the enumeration of k-rooted quadrangulations, k-roote d simple quadrangulations, and k-rooted irreducible quadrangulations." }{TEXT 206 0 "" }}}}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 23 "Method of quotient maps" }{TEXT 207 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 640 "The enumeration of k-rooted unconstrained quadrangulations is eas y to perform using the theory of quotient maps introduced by Liskovets . Cutting the sphere along two meridians forming a diehdral angle of \+ 2pi/k, we extract a sector of the map. Then, pasting the two meridians together, the sector becomes a map having only one root, i.e. a roote d map. In addition, this map has two marked cells, which are the poles of the k-rooted quadrangulation. The quotient map of a k-rooted quadr angulation is a rooted map having only quadrangular faces except in th e case of polar faces (when k=2), which become faces of degree 2 in th e quotient map." }{TEXT 209 0 "" }}}}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 36 "Tree-decomposition by multiple edges" }{TEXT 207 0 "" }} {EXCHG {PARA 267 "" 0 "" {TEXT 209 523 "A tree-decomposition, called \+ \"separating 4-cycle\", allows to find equations linking the generatin g functions of k-rooted irreducible quadrangulations and the generati ng functions of k-rooted simple quadrangulations, which are known from 4). Hence, these equations allow to obtain the generating functions o f k-rooted 3-connected maps, which are also the generating functions o f k-rooted 3-connected maps using Tutte's bijection. Then the enumerat ion of unrooted 3-connected maps is obtained using Burnside-Liskovets \+ lemma." }{TEXT 209 0 "" }}}}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 41 "Tr ee-decomposition by separating 4-cycles" }{TEXT 207 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 523 "A tree-decomposition, called \"separa ting 4-cycle\", allows to find equations linking the generating functi ons of k-rooted irreducible quadrangulations and the generating funct ions of k-rooted simple quadrangulations, which are known from 4). Hen ce, these equations allow to obtain the generating functions of k-root ed 3-connected maps, which are also the generating functions of k-root ed 3-connected maps using Tutte's bijection. Then the enumeration of u nrooted 3-connected maps is obtained using Burnside-Liskovets lemma." }{TEXT 209 0 "" }}}}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 46 "Using alge braicity of series of k-rooted maps " }{TEXT 207 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 579 "The preceding scheme allows to obtain the e numeration of unrooted 2-connected and 3-connected maps. However, one \+ can notice that all series of k-rooted, k-rooted simple and k-rooted \+ irreducible quadrangulations are algebraic because they can be express ed as rational expressions of generating functions of families of tree s. The algebraicity of these generating functions can be used to obtai n fast computation of their coefficient. These methods require to use \+ the package GFUN available on version 9 of maple and downlodable on th e web at http://algo.inria.fr/libraries/down#" }{TEXT 209 0 "" }}}}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory):" } {MPLTEXT 1 0 0 "" }}{PARA 272 "" 1 "" {TEXT 230 68 "Warning, the prote cted name order has been redefined and unprotected" }{TEXT 230 0 "" }} }{EXCHG {PARA 267 "" 0 "" {TEXT 209 113 "The variable N corresponds to the number of edges until which we want to perform the enumeration of unrooted maps" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 6 "N:=20;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"NG6\"\"#?" }{TEXT 20 0 "" }}}{SECT 1 {PARA 268 "" 0 "" {TEXT 208 47 "Enumeration with respect to the number of edges" } {TEXT 208 0 "" }}{SECT 0 {PARA 274 "" 0 "" {TEXT 205 42 "Enumeration o f unrooted unconstrained maps" }{TEXT 205 0 "" }}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 45 "Introduction of the algebraic series beta(x) " } {TEXT 204 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 156 "We introduce \+ the algebraic generating function beta such that all generating series of normal k-rooted quadrangulations will be rational expressions of b eta" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 29 "Eq _beta_x:=-beta+x+3*beta**2;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I*Eq_beta_xG6\",(I%betaGF%!\"\"I\"xGF%\"\"\"*$F'\"\"# \"\"$" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 126 "We ob tain iteratively the development of beta in x (this will be used to fi nd the first values of unrooted maps on the sphere)" }{TEXT 209 0 "" } }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 17 "dev_beta:=0:n:=0:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 13 "while (n " 0 "" {MPLTEXT 1 0 9 "dev_beta;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#+MI\"xG6\"\"\"\"F&\"\"$\"\"#\"#= F'\"$N\"\"\"%\"%M6\"\"&\"&1-\"\"\"'\"&Gi*\"\"(\"'B#Q*\"\")\"(IAQ*\"\"* \")Y()p&*\"#5\"*/qy\"**\"#6\",UNw8/\"\"#7\"-#H0ha5\"\"#8\".+nbAW=\"\"# 9\"/gBhj(= q/&=\"#>\"4I(*\\%y2>-a?\"#?-I\"OGI*protectedGFM6#F&\"#@" }{TEXT 20 0 " " }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 57 "All series of k-rooted q uadrangulations are beta-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 188 "In this section, we obtain beta-rational expres sions for the generating functions of k-rooted quadrangulations. To ob tain these expressions, we use the method of quotient maps of Liskovet s" }{TEXT 223 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 110 "We use t he algebraic equation of beta to find a rational expression in beta fo r beta'(x) which is noted d_beta" }{TEXT 209 0 "" }}{PARA 273 "" 1 "" {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 57 "Eq_diffbe ta_beta_x:=diff(subs(beta=beta(x),Eq_beta_x),x);" }{MPLTEXT 1 0 0 "" } }{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3Eq_diffbeta_beta_xG6\",(-I%diffGI *protectedGF)6$-I%betaGF%6#I\"xGF%F.!\"\"\"\"\"F0*&F+F0F'F0\"\"'" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 81 "d_beta:=s olve(subs(diff(beta(x),x)=dbeta,beta(x)=beta,Eq_diffbeta_beta_x),dbeta );" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I'd_betaG6 \",$*$,&!\"\"\"\"\"I%betaGF%\"\"'F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 204 "We give beta-rational expressions for x, F, f , f' (which is noted d_f), F' (which is noted d_F), F'' (which is n oted d_d_F). The beta-rational expression of F was found in the book o f Goulden and Jackson" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 31 "Exp_x_beta:=solve(Eq_beta_x,x);" }{MPLTEXT 1 0 0 "" } }{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+Exp_x_betaG6\",&I%betaGF%\"\"\"*$ F'\"\"#!\"$" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 38 "F:=factor((1-4*beta)/(1-3*beta)**2-1);" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"FG6\",$*(I%betaGF%\"\"\",&!\"#F)F (\"\"*F),&!\"\"F)F(\"\"$F+F." }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 19 "f:=factor(F/(1+F));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"*(I%betaGF%\"\"\",&!\"#F(F'\"\"*F( ,&!\"\"F(F'\"\"%F-" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 33 "d_f:=factor(diff(f,beta)*d_beta);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$d_fG6\",$*&,&!\"\"\"\"\"I%betaG F%\"\"$F*,&F)F*F+\"\"%!\"#F/" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 33 "d_F:=factor(diff(F,beta)*d_beta);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$d_FG6\",$*$,&!\"\"\"\"\"I %betaGF%\"\"$!\"$!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 271 "> " 0 "" {MPLTEXT 1 0 37 "d_d_F:=factor (diff(d_F,beta)*d_beta);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I&d_d_FG6\",$*&,&!\"\"\"\"\"I%betaGF%\"\"$!\"%,&F)F*F+ \"\"'F)!#=" }{TEXT 20 0 "" }}}{PARA 277 "" 0 "" {XPPMATH 20 "6#>%&d_d_ FG,$*(\"#=\"\"\",&F(!\"\"*&\"\"$F(%%betaGF(F(!\"%,&F(F**&\"\"'F(F-F(F( F*F*" }{TEXT 20 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 1087 "We obt ain beta-rational expressions of k-rooted quadrangulations. The genera ting function Ffv corresponds to the type face-vertex of a 2-rooted qu adrangulation. The generating function Fff corresponds to the type fac e-face of a 2-rooted quadrangulation. The generating function Fvv corr esponds to the type vertex-vertex of a k-rooted quadrangulation . Thes e rational expressions are obtained as linear combinations of derivati ons of F. These linear combinations are easily obtained from the theor y of quotient maps by counting how many times a rooted map can be lift ed in a k-rooted map of type principal cell-principal cell (resp. pri ncipal cell-edge, resp. edge-edge). For example a k-rooted quadrangula tions with kn faces and whose poles are two vertices are in bijection, by taking their quotient map, with rooted quadrangulations with n fac es and two marked vertices. Noting Fvv_n the number of k-rooted quadra ngulations of type vertex-vertex, we have thus Fvv_n=1/2*(n+2)*(n+1)*F _n. Noting Fvv(x)=sum_n Fvv_n x^n and F(x)=sum_n F_n, we have Fvv(x) \+ =1/2 x^2 d^2 f/dx^2 +2x df/dx+F" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 " > " 0 "" {MPLTEXT 1 0 58 "Ffv:=factor(2*d_d_F*Exp_x_beta**2+7*d_F*Exp_ x_beta+2*F+2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6# >I$FfvG6\",$*&,&!\"\"\"\"\"I%betaGF%\"\"'F),&F)F*F+\"\"$F)\"\"#" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 56 "Fff:=fact or(2*d_d_F*Exp_x_beta**2+5*d_F*Exp_x_beta+F+1);" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I$FffG6\",$*&,&!\"\"\"\"\"I%betaGF% \"\"$!\"#,&F)F*F+\"\"'F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 56 "Fvv:=factor(1/2*d_d_F*Exp_x_beta**2+2*d_F*Exp_x_be ta+F);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$FvvG6 \",$*&I%betaGF%\"\"\",&!\"\"F)F(\"\"'F+!\"'" }{TEXT 20 0 "" }}}{PARA 267 "" 0 "" {TEXT 209 0 "" }}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 72 " Obtaining the first N coefficients counting unrooted maps on the sphe re" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 281 "In this \+ section, we obtain the first N terms counting unrooted maps on the sp here by their number of edges. To do this, we obtain first the develop ment of the series of k-rooted quadrangulations. Then, from Burnside l emma, we obtain the enumeration of unrooted maps on the sphere" } {TEXT 223 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 230 "We obtain, f rom the development of beta, developments of the generating functions \+ of k-rooted quadrangulations. These developments will be used to have \+ the first values of non-rooted planar maps with respect to the number \+ of edges" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 43 "dev_F:=series(subs(beta=dev_beta,F),x,N+1);" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I&dev_FG6\"+MI\"xGF%\"\"#\"\"\"\"\"* F(\"#a\"\"$\"$y$\"\"%\"%;H\"\"&\"&dS#\"\"'\"'%\\3#\"\"(\"(Yk(=\"\")\") s(*R(=q/&=\"#=\" 3gUruS8@c>\"#>\"4?i]wp^(p!3#\"#?-I\"OGI*protectedGFP6#F)\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 47 "dev_Fvv:=series (subs(beta=dev_beta,Fvv),x,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I(dev_FvvG6\"+MI\"xGF%\"\"'\"\"\"\"#a\"\"#\"$S&\"\" $\"%qc\"\"%\"&O7'\"\"&\"''ft'F(\"(%y0v\"\"(\")q+W%)\"\")\"*gu)p&*\"\"* \",Wql44\"\"#5\"-/D;l\\7\"#6\".'zo$*4P9\"#7\"/+Qz:>e;\"#8\"0+a=ak(=>\" #9\"1S1boowDA\"#:\"2!>,p%)RX(e#\"#;\"3gQhGwGi8I\"#<\"4+<;ncN$*e^$\"#= \"5+Y***ob\"Q/3T\"#>\"6?o$Hd@k7T1[\"#?-I\"OGI*protectedGFQ6#F)\"#@" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 47 "dev_Ffv:= series(subs(beta=dev_beta,Ffv),x,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I(dev_FfvG6\"+OI\"xGF%\"\"#\"\"!\"#=\"\"\" \"$!=F(\"%!*=\"\"$\"&7/#\"\"%\"'KXA\"\"&\"(G>]#\"\"'\")!pY\"G\"\"(\"*? e**=$\"\")\"+[BbOO\"\"*\",oT0b;%\"#5\"-KH7L!z%\"#6\".+YE0t_&\"#7\"/+=1 =)eR'\"#8\"0!)o^*GA>u\"#9\"1IPj:m%[i)\"#:\"3?Y?we4a/5\"#;\"4+R0*)=Xk>< \"\"#<\"5+#)**H_QzMp8F*\"6S*y4>2)3P@g\"\"#>\"7S/h3GJg\"*yw=\"#?-I\"OGI *protectedGFR6#F+\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 47 "dev_Fff:=series(subs(beta=dev_beta,Fff),x,N+1);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I(dev_FffG6\"+OI \"xGF%\"\"\"\"\"!\"#7F(\"$N\"\"\"#\"%7:\"\"$\"&5q\"\"\"%\"'cC>\"\"&\"( (=*=#\"\"'\")!G>]#\"\"(\"*Qi4(G\"\")\"+!ocfI$\"\"*\",a'*z$=Q\"#5\"-o6U %=U%\"#6\".+du(\\K^F*\"/!odo*[pf\"#8\"0v?UY@b&p\"#9\"1Sq1(p-v6)\"#:\"2 I5(>x7L([*\"#;\"4+e%fJ7@G56\"#<\"5!H)\\ofT0)3I\"\"#=\"6+G7x#yX[%e_\"\" #>\"7?a&=x;@c![\"z\"\"#?-I\"OGI*protectedGFR6#F(\"#@" }{TEXT 20 0 "" } }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 139 "We perform the sum of coeffi cients of the series of k-rooted maps as indicated by Burnside Lemma f or maps on the sphere (Liskovets formula)" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 125 "sum_k_rooted:=series(dev_F+x*sub s(x=x**2,dev_Ffv)+x**2*subs(x=x**2,dev_Fff)+sum(phi(k)*subs(x=x**k,dev _Fvv),k=2..N+1),x,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I-sum_k_rootedG6\"+MI\"xGF%\"\"%\"\"\"\"#;\"\"#\"#%)\" \"$\"$c%F(\"%?J\"\"&\"&_[#\"\"'\"'?/@\"\"(\"(gP)=\"\")\")+8U<\"\"*\"*? jPl\"\"#5\"+g?M-;\"#6\",g$zJz:\"#7\"-g&4b#z:\"#8\".wo9!)*)f\"\"#9\"/g$ G6'fO;\"#:\"0%=$**[Z6p\"F*\"1wvAyGNi<\"#<\"2gZO \"#>\"4g`E#>Jvp!3#\"#?-I\"OGI*protectedGFP6#F)\"#@" }{TEXT 20 0 "" }}} {EXCHG {PARA 267 "" 0 "" {TEXT 209 174 "We divide each coefficient of \+ the preceding sum by 2*n to achieve the Burnside Lemma and obtain the \+ first values of non-rooted planar maps with respect to the number of e dges" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "u nrooted_maps:=series(1/2*int(series(sum_k_rooted/x,x,N+1),x),x,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.unrooted_maps G6\"+MI\"xGF%\"\"#\"\"\"\"\"%F(\"#9\"\"$\"#dF*\"$7$\"\"&\"%r?\"\"'\"&I ]\"\"\"(\"'Nx6\"\")\"']y'*\"\"*\"(;)o#)\"#5\")IP$G(\"#6\"*S\"\\!e'\"#7 \"+g!eS2'\"#8\",S^\"#=\"1#G7mnCz9&\"#>\"2Mm!)z#Qu,_\"#?-I\"OGI*protectedGFO6#F) \"#@" }{TEXT 20 0 "" }}}}}{PARA 267 "" 0 "" {TEXT 209 0 "" }}{SECT 0 {PARA 274 "" 0 "" {TEXT 205 40 "Enumeration of unrooted 2-connected ma ps" }{TEXT 205 0 "" }}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 93 "All seri es of k-rooted simple quadrangulations composed with x->y=x*(1+F)^2 ar e beta-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 237 "From the equations given by the tree decomposition by multiple ed ges, we obtain rational expressions in terms of beta(x) for the genera ting functions of k-rooted simple quadrangulations composed with the c hange of variable x->y=x*(1+F)^2" }{TEXT 223 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 220 "We obtain a rational expression of the generati ng function E which is useful in the equations linking generating func tions of k-rooted normal quadrangulations and generating functions of \+ k-rooted simple quadrangulations " }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 32 "E:=factor(2*Exp_x_beta*d_F+F+1);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"EG6\"*$,&!\" \"\"\"\"I%betaGF%\"\"$!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 431 "From the equations linking k-rooted normal quadrangula tions with k-rooted simple quadrangulations, we obtain rational expres sions of the generating series of k-rooted simple quadrangulations in \+ terms of beta. gvv, gfv and gff denote respectively the generating fun ctions of k-rooted simple quadrangulations of type vertex-vertex, of 2 -rooted quadrangulations of type face-vertex, and of 2-rooted quadrang ulations of type face-face." }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 52 "gvv:=factor((Fvv-2*Exp_x_beta*d_f*(1+F))/(E*(1-f )));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gvvG6\" ,$*&I%betaGF%\"\"\",&!\"\"F)F(\"\"'F+!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 19 "gfv:=factor(Ffv/E);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gfvG6\",$*&,&!\"\"\"\"\"I %betaGF%\"\"'F),&F)F*F+\"\"$F*\"\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 25 "gff:=factor(Fff*(1-f)/E);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gffG6\"*(,&!\"\"\"\"\"I%b etaGF%\"\"$\"\"#,&F(F)F*\"\"'F(,&F(F)F*\"\"%F(" }{TEXT 20 0 "" }}} {PARA 277 "" 0 "" {TEXT 20 0 "" }}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 43 "Introduction of the algebraic series eta(y)" }{TEXT 204 0 "" } }{EXCHG {PARA 276 "" 0 "" {TEXT 223 173 "In this section, we introduce the algebraic series eta(y) and obtain beta(x) as a rational expressi on of eta(y) when y and x are linked by the change of variable y=x*(1+ F)^2" }{TEXT 223 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 183 "We ma ke a change of variable by introducing the variable y such that y=x*(1 +F)**2. This variable will be the variable for the generating series o f the k-rooted simple quadrangulations" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 46 "Eq_y_F:=numer(factor(-y+Exp_x_bet a*(1+F)**2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#> I'Eq_y_FG6\",0I\"yGF%\"\"\"*&F'F(I%betaGF%F(!\"**&F'F(F*\"\"#\"#F*&F'F (F*\"\"$!#FF*!\"\"*$F*F-\"\")*$F*F0!#;" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 143 "We introduce the algebraic sery eta s uch that all the generating series of k-rooted simple quadrangulations will be rational expressions of eta" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 28 "Eq_y_eta:=-eta*(1-eta)**2+y;" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I)Eq_y_etaG6\",& *&I$etaGF%\"\"\",&F)F)F(!\"\"\"\"#F+I\"yGF%F)" }{TEXT 20 0 "" }}} {EXCHG {PARA 267 "" 0 "" {TEXT 209 123 "We obtain iteratively the deve lopment of eta in y (this will be used to find the first values of unr ooted 2-connected maps)" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 16 "dev_eta:=0:n:=0:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 13 "while(n " 0 "" {MPLTEXT 1 0 8 "dev_eta;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#+MI\"yG6\"\"\"\"F&\"\"#F'\"\"(\"\"$\"#I\"\"%\"$V\"\"\"& \"$G(\"\"'\"%wQF(\"&=8#\"\")\"'v,7\"\"*\"'!p!p\"#5\"(:?.%\"#6\")![TQ# \"#7\"*#p)\\U\"\"#8\"*?f^f)\"#9\"+CSED_\"#:\",MDn$)>$\"#;\"-ByeZp>\"#< \".!>`$*>>7\"#=\".D>\\UJe(\"#>\"/q=kuaOZ\"#?-I\"OGI*protectedGFM6#F&\" #@" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 26 "Exp_ y_eta:=eta*(1-eta)**2;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I*Exp_y_etaG6\"*&I$etaGF%\"\"\",&F(F(F'!\"\"\"\"#" } {TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 276 "We have the i ntuition that beta(x) must be a rational expression of eta(y) (if thi s is the case, all generating functions of simple quadrangulations com posed with y=x*(1+F)^2, which are rational expressions of beta(x), wil l be rational expressions of eta(y), which we expect)" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 50 "Eq_eta_beta:=factor(re sultant(Eq_y_F,Eq_y_eta,y));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I,Eq_eta_betaG6\"*&,(*&I$etaGF%\"\"\"I%betaGF%F*\"\"$F +F*F)!\"\"F*,4*$F)\"\"#F**&F)F0F+F*!\"'*&F)F0F+F0\"\"**&F)F*F+F0!#@F( \"#8F)!\"#F*F*F+!\")*$F+F0\"#;F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 " " 0 "" {TEXT 209 125 "We conjecture that the first factor is always nu ll, and thus conjecture from it a rational expression of beta in terms of eta" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 46 "Exp_beta_eta:=solve(3*eta*beta+beta-eta,beta);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I-Exp_beta_etaG6\"*&I$etaGF%\"\" \",&F'\"\"$F(F(!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 234 "We check that this rational expression is correct by ch ecking that the following expression is null (in fact we check that th e relation y=x*(1+F)**2 remains true when we replace y, x and F by the ir expression in terms in terms of eta)" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 63 "factor(subs(beta=eta/(3*eta+1),Ex p_x_beta*(1+F)**2)-Exp_y_eta);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 63 "All series of k-rooted simple quadrangulations are eta-r ational" }{TEXT 204 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 396 "By \+ substituting the expression of beta in terms of eta in the rational ex pressions of k-rooted simple quadrangulations in terms of beta, we obt ain rational expressions of k-rooted simple quadrangulations in terms \+ of eta. We also obtain a rational expression of the generating functio n of rooted simple quadrangulations in terms of eta (this expression i s given in the book of Goulden and Jackson)" }{TEXT 209 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "gvv:=factor(subs(beta=Exp_ beta_eta,gvv));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6 #>I$gvvG6\",$*&I$etaGF%\"\"\",&F(\"\"$!\"\"F)F,!\"#" }{TEXT 20 0 "" }} }{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "gfv:=factor(subs(beta=Exp _beta_eta,gfv));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 " 6#>I$gfvG6\",$*$,&I$etaGF%\"\"$!\"\"\"\"\"F+!\"#" }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "gff:=factor(subs(beta=Exp_ beta_eta,gff));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6 #>I$gffG6\"*&,&!\"\"\"\"\"I$etaGF%F)F(,&F*\"\"$F(F)F(" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 83 "Obtaining the first N coef ficients counting unrooted 2-connected maps on the sphere" }{TEXT 204 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 248 "We obtain, from the dev elopment of eta, developments of the generating functions of k-rooted \+ simple quadrangulations. These developments will be used to have the f irst values of non-rooted 2-coonected planar maps with respect to the \+ number of edges" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 45 "dev_gvv:=series(subs(eta=dev_eta,gvv),y,N+1);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I(dev_gvvG6\"+MI \"yGF%\"\"#\"\"\"\"#5F(\"#c\"\"$\"$I$\"\"%\"%-?\"\"&\"&wB\"\"\"'\"&?v( \"\"(\"'9.\\\"\")\"(]X7$\"\"*\")5+.?F*\"*![C!H\"\"#6\"*+=XM)\"#7\"+'H] \\T&\"#8\",?F:S_$\"#9\"-cqh6*H#\"#:\".)44EB.:\"#;\".]6Rzt%)*\"#<\"/q!> dc\"1I.(Q+jXz#\"#?-I\"OGI*protectedGFP6#F)\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 45 "dev_gfv: =series(subs(eta=dev_eta,gfv),y,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I(dev_gfvG6\"+OI\"yGF%\"\"#\"\"!\"\"'\"\"\"\"# IF(\"$o\"\"\"$\"$!**\"\"%\"%1g\"\"&\"&Gr$F*\"'gDB\"\"(\"(U4Z\"\"\")\"( ]OP*\"\"*\")I+4g\"#5\"*SM2(Q\"#6\"++aN.D\"#7\",))3&[C;\"#8\"-g\"e/s0\" \"#9\"-o6&[t*o\"#:\".%HFyp4X\"#;\"/]M\"#=\"1+MkQz' RF\"\"#>\"1!*4h6!*o$Q)\"#?-I\"OGI*protectedGFR6#F+\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 45 "dev_gff:=series(subs(e ta=dev_eta,gff),y,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I(dev_gffG6\"+OI\"yGF%\"\"\"\"\"!\"\"%F(\"#@\"\"#\"$? \"\"\"$\"$:(F*\"%oV\"\"&\"&Kr#\"\"'\"'W0<\"\"(\"(v:3\"\"\")\"(+p!p\"\" *\")l@NW\"#5\"*gx4'G\"#6\"+'*H[_=\"#7\",!)GAL?\"\"#8\",g.'*y$y\"#9\"-W 0wQ<^\"#:\".\"*H**3\"[L\"#;\"/?uN)eX>#\"#<\"0vlM2(zS9\"#=\"0+u$G\\4t%* \"#>\"1&RfPqYmB'\"#?-I\"OGI*protectedGFR6#F(F+" }{TEXT 20 0 "" }}} {EXCHG {PARA 267 "" 0 "" {TEXT 209 301 "To apply Burnside lemma to enu merate unrooted 2-connected maps, we also need the generating function g(y) of rooted simple quadrangulations whose eta-rational expression \+ is for example given in the article of Mullin and Schellenberg. From t his eta-rational expression, we obtain a development for g(y)" }{TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 17 "g:=eta*(2-3*et a);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"gG6\"*& I$etaGF%\"\"\",&\"\"#F(F'!\"$F(" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "dev_g:=series(subs(eta=dev_eta,g),y,N+1);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I&dev_gG6\"+MI\" yGF%\"\"#\"\"\"F)F(F(\"\"$\"\"'\"\"%\"#A\"\"&\"#\"*F+\"$3%\"\"(\"%Q>\" \")\"%9'*\"\"*\"&N$\\\"#5\"'I,E\"#6\"(SCS\"\"#7\"(KEq(\"#8\")'zvH%\"#9 \"*Ob.V#\"#:\"+eWf!R\"\"#;\"+aqnQ!)\"#<\",:GG#*o%\"#=\"-qgj]dF\"#>\".I !HAHL;\"#?-I\"OGI*protectedGFN6#F)\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 223 "We perform the sum of coefficients of the series of k-rooted 2-connected maps (which are in bijection via t he Tutte's application with the k-rooted simple quadrangulations) as i ndicated by the Burnside Lemma for planar maps" }{TEXT 209 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 134 "sum_k_rooted_2connected:= series(dev_g+y*subs(y=y**2,dev_gfv)+y**2*subs(y=y**2,dev_gff)+sum(phi( k)*subs(y=y**k,dev_gvv),k=2..N),y,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I8sum_k_rooted_2connectedG6\"+MI\"yGF%\"\" %\"\"\"F(\"\"#\"#7\"\"$\"#CF(\"#g\"\"&\"$#>\"\"'\"$)e\"\"(\"%;C\"\")\" &G2\"\"\"*\"&+@&\"#5\"'chE\"#6\"(%)*>9F+\"(%yRx\"#8\")?03V\"#9\"*SBFV# \"#:\"+KgD\"R\"\"#;\"+G![,/)\"#<\",wQ^'*o%\"#=\"-c(4+wv#\"#>\".?8<>Lj \"\"#?-I\"OGI*protectedGFO6#F)\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 179 "We divide each coefficient of the preceding sum by 2*n to achieve the Burnside Lemma and obtain the first values \+ of non-rooted 2-connected maps with respect to the number of edges" } {TEXT 209 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "unrooted _2connected_maps:=series(1/2*int(series(sum_k_rooted_2connected/y,y,N+ 1),y),y,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#> I9unrooted_2connected_mapsG6\"+MI\"yGF%\"\"#\"\"\"F)F(F(\"\"$F*\"\"%\" \"'\"\"&\"#;F,\"#U\"\"(\"$^\"\"\")\"$'f\"\"*\"%0E\"#5\"&)47\"#6\"&m\"f \"#7\"'%o(H\"#8\"(!fQ:\"#9\"(y!4\")\"#:\")^nZVF.\"*U\\ZO#\"#<\"+T4o-8 \"#=\"+iB%oD(\"#>\",$GzH$3%\"#?-I\"OGI*protectedGFL6#F)\"#@" }{TEXT 20 0 "" }}}}}{SECT 0 {PARA 274 "" 0 "" {TEXT 205 40 "Enumeration of un rooted 3-connected maps" }{TEXT 205 0 "" }}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 142 "Obtaining eta-rational expressions for series linked to simple quadrangulations and appearing in the equations given by the t ree-decomposition" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 249 "In this section, we introduce all generating functions of bi- rooted, rooted or k-rooted simple quadrangulations which will appear i n the equations linking series of k-rooted irreducible quadrangulation s and series of k-rooted simple quadrangulations" }{TEXT 223 0 "" }}} {EXCHG {PARA 270 "" 0 "" {TEXT 206 87 "We derivate the algebraic equat ion of eta to find an eta-rational expression of d_beta." }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 52 "Eq_diffeta_eta_y:= diff(subs(eta=eta(y),Eq_y_eta),y);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I1Eq_diffeta_eta_yG6\",(*&-I%diffGI*protectedGF*6 $-I$etaGF%6#I\"yGF%F/\"\"\",&F0F0F,!\"\"\"\"#F2*(F,F0F1F0F(F0F3F0F0" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 83 "d_eta:=fa ctor(solve(subs(diff(eta(y),y)=d_eta,eta(y)=eta,Eq_diffeta_eta_y),d_et a));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I&d_etaG6 \"*&,&!\"\"\"\"\"I$etaGF%F)F(,&F*\"\"$F(F)F(" }{TEXT 20 0 "" }}} {EXCHG {PARA 267 "" 0 "" {TEXT 209 218 "W(y) corresponds to the genera ting function g(y)-2*y (it counts the rooted simple quadrangulations w ith respect to the number of faces without counting the two degenerate d rooted simple quadrangulations with two edges)" }{TEXT 209 0 "" }}} {EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 22 "W:=(eta-2*eta**2)*eta;" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"WG6\"*&,&I $etaGF%\"\"\"*$F(\"\"#!\"#F)F(F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 " " 0 "" {TEXT 209 231 "We obtain a rational expression of the generatin g function C which will be useful in the equations linking generating \+ functions of k-rooted simple quadrangulations and generating functions of k-rooted irreductible quadrangulations " }{TEXT 209 0 "" }}} {EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 41 "C:=factor(Exp_y_eta*diff (W,eta)*d_eta-W);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"CG6\"*$I$etaGF%\"\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 152 "G(y) corresponds to the generating function of sim ple rooted quadrangulations such that the root node of the decompositi on tree isn't an horizontal axis" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 29 "G:=factor(W/(1+W/Exp_y_eta));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"GG6\"**I$etaGF%\"\"# ,&!\"\"\"\"\"F'F(F+,&F*F+F'F+F(,(F*F+F'F+*$F'F(F+F*" }{TEXT 20 0 "" }} }{EXCHG {PARA 267 "" 0 "" {TEXT 209 236 "We obtain also a rational exp ression of the generating function B which will be useful in the equat ions linking generating functions of k-rooted simple quadrangulations \+ and generating functions of k-rooted irreductible quadrangulations " } {TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 41 "B:=fac tor(Exp_y_eta*diff(G,eta)*d_eta-G);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"BG6\"*(,&!\"\"\"\"\"I$etaGF%F)\"\"%F*\"\"#, (F(F)F*F)*$F*F,F)!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 190 "We obtain by integration an eta-rational expression of \+ g_rootedFace_nonRootedFace. By taking the difference with gff, we also find an eta-rational expression of g_nonRootedFace_nonRootedFace" } {TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 72 "g_root edFace_nonRootedFace_order2:=factor(int(gff/d_eta,eta)/Exp_y_eta);" } {MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IBg_rootedFace _nonRootedFace_order2G6\"*$,&!\"\"\"\"\"I$etaGF%F)!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 84 "g_nonRootedFace_nonR ootedFace_order2:=factor(gff-g_rootedFace_nonRootedFace_order2);" } {MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IEg_nonRootedF ace_nonRootedFace_order2G6\",$*(I$etaGF%\"\"\",&!\"\"F)F(F)!\"#,&F(\" \"$F+F)F+F," }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 287 "After a small calcul made by hand (it is a little tricky because we m ust introduce the algebraic generating function eta2=y**2/(1-eta2)**2) , we also find this rational eta-expression for the 2-rooted simple q uadrangulations with one axial face whose 2-root is incident to the ax ial face" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 42 "g_rootedFace_vertex_order2:=2*eta/(1-eta);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I;g_rootedFace_vertex_order2G6 \",$*&I$etaGF%\"\"\",&F)F)F(!\"\"F+\"\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 93 "We check that the generating function \+ of g_rootedFace_vertice_order2 found by hand is correct" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 98 "factor(2*Exp_y_eta *diff(g_rootedFace_vertex_order2,eta)*d_eta+g_rootedFace_vertex_order2 -(gfv-2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\" \"!" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 107 "By taki ng the difference with gfv, we also find an eta-rational expression of g_nonRootedFace_vertex_order2" }{TEXT 223 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 72 "g_nonRootedFace_vertex_order2:=factor(gfv-2 -g_rootedFace_vertex_order2);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 " " {XPPMATH 20 "6#>I>g_nonRootedFace_vertex_order2G6\",$*(I$etaGF%\"\" \",&!\"\"F)F(F)F+,&F(\"\"$F+F)F+\"\"%" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 141 "We introduce the two adjoint functions h an d l which will be useful to obtain respectively q_rootedFace_nonRooted Face and q_rootedFace_vertex" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 61 "l:=factor(g_rootedFace_nonRootedFace_order2*( 1-G/Exp_y_eta));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"lG6\",$*$,(!\"\"\"\"\"I$etaGF%F**$F+\"\"#F*F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 68 "h:=factor((g_roo tedFace_vertex_order2-W/Exp_y_eta)*(1-G/Exp_y_eta));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"hG6\",$*&I$etaGF%\"\"\",(! \"\"F)F(F)*$F(\"\"#F)F+F+" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 " " {TEXT 204 95 "All series of k-rooted irreducible quadrangulations c omposed with y->z=W(y)/y are eta-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 244 "From the equations given by the tree \+ decomposition by separating 4-cycles, we obtain rational expressions i n terms of eta(y) for the generating functions of k-rooted irreducible quadrangulations composed with the change of variable y->z=W(y)/y" } {TEXT 223 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 63 "qvv_or derNot2:=factor((gvv-2*B/Exp_y_eta/(1-G/Exp_y_eta))*W/C);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.qvv_orderNot2G6\",$*, ,&!\"\"\"\"\"I$etaGF%\"\"#F*,&F+F*!\"#F*F*F+F,,(F)F*F+F**$F+F,F*F),&F+ \"\"$F)F*F)F," }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 97 "q_rootedFace_nonRootedFace_order2:=factor((l-l*W/Exp_y_eta-1 )/g_rootedFace_nonRootedFace_order2);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IBq_rootedFace_nonRootedFace_order2G6\",$* (I$etaGF%\"\"#,(\"\"\"F+F(!\"\"*$F(F)F+F+,(F,F+F(F+F-F+F,F," }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 117 "q_rootedFace _vertex_order2:=factor(h-(h+1)*W/Exp_y_eta-q_rootedFace_nonRootedFace_ order2*g_rootedFace_vertex_order2);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I;q_rootedFace_vertex_order2G6\",$*&I$etaGF%\" \"%,(!\"\"\"\"\"F(F,*$F(\"\"#F,F+!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 147 "q_nonRootedFace_nonRootedFace_order2 :=factor((g_nonRootedFace_nonRootedFace_order2-2*C/Exp_y_eta*l**2)*W/( C*g_rootedFace_nonRootedFace_order2**2));" }{MPLTEXT 1 203 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>IEq_nonRootedFace_nonRootedFace_orde r2G6\",$*.,&!\"\"\"\"\"I$etaGF%F*\"\"#,&F)F*F+F,F*,*F*F*F+F)*$F+F,F,*$ F+\"\"$F*F*F+F,,(F)F*F+F*F/F*!\"#,&F+F1F)F*F)F," }{TEXT 20 0 "" }}} {EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 237 "q_nonRootedFace_vertex_ order2:=factor((g_nonRootedFace_vertex_order2-4*C/Exp_y_eta*l*(1+h)-2* C/W*q_nonRootedFace_nonRootedFace_order2*g_rootedFace_nonRootedFace_or der2*g_rootedFace_vertex_order2)*W/(C*g_rootedFace_nonRootedFace_order 2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I>q_non RootedFace_vertex_order2G6\",$*.I$etaGF%\"\"%,&!\"\"\"\"\"F(\"\"#F,,(* $F(F-F-F(\"\"$!\"%F,F,,&F+F,F(F,F,,(F+F,F(F,F/F,!\"#,&F(F0F+F,F+F)" } {TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 239 "q_vert ex_vertex_order2:=factor((gvv-2*B/Exp_y_eta/(1-G/Exp_y_eta)-4*C/Exp_y_ eta*h-2*C/Exp_y_eta*h**2-C/W*q_nonRootedFace_nonRootedFace_order2*g_ro otedFace_vertex_order2**2-C/W*q_nonRootedFace_vertex_order2*g_rootedFa ce_vertex_order2)*W/C);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I7q_vertex_vertex_order2G6\",$*.I$etaGF%\"\"%,&!\"\"\" \"\"F(\"\"#F,,.*$F(\"\"&F)*$F(F)!\"%*$F(\"\"$!#:*$F(F-\"#@F(!#5F-F,F,, &F+F,F(F,!\"#,&F(F4F+F,F+,(F+F,F(F,F6F,F:F-" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 45 "Introduction of the algebraic series gamma(z)" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 173 "I n this section, we introduce the algebraic series gamma(z) and obtain \+ eta(y) as a rational expression of gamma(z) when z and y are linked by the change of variable z=W(y)/y" }{TEXT 223 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 185 "We make a change of variable by introducing the variable z such that z=W(y)/y. This variable will be the variable for the generating series of the k-rooted irreductible quadrangulations" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 38 "Eq_z_ W:=factor(numer(-z+W/Exp_y_eta));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I'Eq_z_WG6\",,I\"zGF%!\"\"*&F'\"\"\"I$etaGF%F*\" \"#*&F'F*F+F,F(F+F**$F+F,!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 168 "We introduce the algebraic generating function gamm a such that all the generating series of k-rooted irreductible quadran gulations will be rational expressions of gamma" }{TEXT 209 0 "" }}} {EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 34 "Eq_z_gamma:=-gamma+z*(1+ gamma)**2;" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I +Eq_z_gammaG6\",&I&gammaGI*protectedGF(!\"\"*&I\"zGF%\"\"\",&F,F,F'F, \"\"#F," }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 32 "Exp_z_gamma:=gamma/(1+gamma)**2;" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I,Exp_z_gammaG6\"*&I&gammaGI*protectedGF( \"\"\",&F)F)F'F)!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 241 "We have the intuition that eta must be a rational expre ssion of gamma (if this is the case all generating series of irreducti ble quadrangulations, which are rational expressions of eta, will be r ational expressions of gamma, which we expect)" }{TEXT 209 0 "" }}} {EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 53 "Eq_eta_gamma:=factor(res ultant(Eq_z_W,Eq_z_gamma,z));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 " " {XPPMATH 20 "6#>I-Eq_eta_gammaG6\"*&,(*&I&gammaGI*protectedGF*\"\"\" I$etaGF%F+F+!\"\"F+F,\"\"#F+,(F)F-F(F.F,F+F+" }{TEXT 20 0 "" }}} {EXCHG {PARA 267 "" 0 "" {TEXT 209 127 "We conjecture that the second \+ factor is always null, and thus conjecture from it a rational expressi on of eta in terms of gamma" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 49 "Exp_eta_gamma:=solve(-gamma+2*gamma*eta+eta,et a);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.Exp_et a_gammaG6\"*&I&gammaGI*protectedGF(\"\"\",&F'\"\"#F)F)!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 49 "We check that this r ational expression is correct" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> \+ " 0 "" {MPLTEXT 1 203 60 "factor(subs(eta=gamma/(2*gamma+1),W/Exp_y_et a-Exp_z_gamma));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 133 "We obtain iteratively a development of gamma. This development will b e useful to find the first values of non rooted 3-connected maps" } {TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 18 "dev_ga mma:=0:n:=0:" }{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 13 "while(n " 0 "" {MPLTEXT 1 203 10 "dev_gamma;" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 " " {XPPMATH 20 "6#+MI\"zG6\"\"\"\"F&\"\"#F'\"\"&\"\"$\"#9\"\"%\"#UF(\"$ K\"\"\"'\"$H%\"\"(\"%I9\"\")\"%i[\"\"*\"&'z;\"#5\"&'ye\"#6\"'7!3#\"#7 \"'+Hu\"#8\"(SWn#F*\"(X[p*\"#:\")qwNN\"#;\"*!zW'H\"\"#<\"*+(QwZ\"#=\"+ !>jsw\"\"#>\"+?/7kl\"#?-I\"OGI*protectedGFL6#F&\"#@" }{TEXT 20 0 "" }} }}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 70 "All series of k-rooted irred ucible quadrangulations are gamma-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 209 427 "By substituting the expression of eta in terms of gamma in the rational expressions of k-rooted irreductibl e quadrangulations in terms of eta, we obtain rational expressions of \+ k-rooted irreductible quadrangulations in terms of gamma. We also obta in a rational expression of the generating function q of rooted irredu ctible quadrangulations in terms of gamma (this expression is given in the article of Mullin and Schellenberg)" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 65 "qvv_orderNot2:=factor(subs(eta= gamma/(2*gamma+1),qvv_orderNot2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 " " 1 "" {XPPMATH 20 "6#>I.qvv_orderNot2G6\",$*,I&gammaGI*protectedGF)\" \"#,&F(\"\"$F*\"\"\"F-,&F(F-!\"\"F-F/,(F-F-F(F,*$F(F*F-F/,&F(F*F-F-F/! \"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 105 " q_rootedFace_nonRootedFace_order2:=factor(subs(eta=gamma/(2*gamma+1),q _rootedFace_nonRootedFace_order2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IBq_rootedFace_nonRootedFace_order2G6\"**,(\" \"\"F(I&gammaGI*protectedGF*\"\"$*$F)\"\"#F+F(F)F-,(F(F(F)F+F,F(!\"\", &F)F-F(F(!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 91 "q_rootedFace_vertex_order2:=factor(subs(eta=gamma/(2*gamma+1 ),q_rootedFace_vertex_order2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I;q_rootedFace_vertex_order2G6\",$*(I&gammaGI*pro tectedGF)\"\"%,(\"\"\"F,F(\"\"$*$F(\"\"#F,!\"\",&F(F/F,F,!\"#F/" } {TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 111 "q_nonR ootedFace_nonRootedFace_order2:=factor(subs(eta=gamma/(2*gamma+1),q_no nRootedFace_nonRootedFace_order2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IEq_nonRootedFace_nonRootedFace_order2G6\",$*. I&gammaGI*protectedGF)\"\"#,*\"\"\"F,F(\"\"&*$F(F*\"#5*$F(\"\"$\"\"*F, ,&F,F,F(F,F*,&F(F,!\"\"F,F5,(F,F,F(F1F.F,!\"#,&F(F*F,F,!\"$F7" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 97 "q_nonRootedFa ce_vertex_order2:=factor(subs(eta=gamma/(2*gamma+1),q_nonRootedFace_ve rtex_order2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 " 6#>I>q_nonRootedFace_vertex_order2G6\",$*.,&\"\"\"F)I&gammaGI*protecte dGF+F)F),(*$F*\"\"#\"\")F*\"#8\"\"%F)F)F*F1,&F*F)!\"\"F)F3,(F)F)F*\"\" $F-F)!\"#,&F*F.F)F)!\"$!\"%" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 83 "q_vertex_vertex_order2:=factor(subs(eta=gamma/( 2*gamma+1),q_vertex_vertex_order2));" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I7q_vertex_vertex_order2G6\",$*.,.*$I&gamm aGI*protectedGF+\"\"&\"\")*$F*\"\"%\"#G*$F*\"\"$\"#J*$F*\"\"#\"#@F*\"# 5F5\"\"\"F8F*F/,(F8F8F*F2F4F8!\"#,&F*F8!\"\"F8F<,&F8F8F*F8F:,&F*F5F8F8 !\"$F:" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 102 "q := -gamma^6*(2*gamma^3-1-4*gamma-3*gamma^2+gamma^4)/(1+gamma)^4 /(1+3*gamma+gamma^2)^2/(2*gamma+1)^3;" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"qG6\",$*,I&gammaGI*protectedGF)\"\"',,* $F(\"\"$\"\"#!\"\"\"\"\"F(!\"%*$F(F.!\"$*$F(\"\"%F0F0,&F0F0F(F0F1,(F0F 0F(F-F2F0!\"#,&F(F.F0F0F3F/" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 84 "Obtaining the N first coefficients counting unroote d 3-connected maps on the sphere" }{TEXT 204 0 "" }}{EXCHG {PARA 267 " " 0 "" {TEXT 209 256 "We obtain, from the development of gamma, develo pments of the generating functions of k-rooted irreductible quadrangul ations. These developments will be used to have the first values of no n-rooted 3-coonected planar maps with respect to the number of edges" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 109 "dev_ q_rootedFace_nonRootedFace_order2:=series(subs(gamma=dev_gamma,q_roote dFace_nonRootedFace_order2),z,N+1);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IFdev_q_rootedFace_nonRootedFace_order2G6\"+II \"zGF%\"\"\"\"\"#\"\"%F*\"\"'\"\"&\"#DF+\"#s\"\"(\"$W#\"\")\"$5)\"\"* \"%wF\"#5\"%='*\"#6\"&yP$\"#7\"'E)>\"\"#8\"'0*G%\"#9\"(opa\"\"#:\"(Sph &\"#;\")qX^?\"#<\"))y9`(\"#=\"*%f&yx#\"#>\"+-J$)G5\"#?-I\"OGI*protecte dGFL6#F(\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 95 "dev_q_rootedFace_vertex_order2:=series(subs(gamma=dev_gamma,q_ rootedFace_vertex_order2),z,N+1);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I?dev_q_rootedFace_vertex_order2G6\"+GI\"zGF%\" \"#\"\"%F(\"\"&\"#7\"\"'\"#K\"\"(\"$9\"\"\")\"$y$\"\"*\"%58\"#5\"%iX\" #6\"&1h\"F+\"&mt&\"#8\"'_g?\"#9\"'w`u\"#:\"(9Lr#\"#;\"(a<$**\"#<\")yM` O\"#=\"*Q0)\\8\"#>\"*Q$)p+&\"#?-I\"OGI*protectedGFJ6#\"\"\"\"#@" } {TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 115 "dev_q_ nonRootedFace_nonRootedFace_order2:=series(subs(gamma=dev_gamma,q_nonR ootedFace_nonRootedFace_order2),z,N+1);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IIdev_q_nonRootedFace_nonRootedFace_order2 G6\"+II\"zGF%\"\"#F(\"#;\"\"%\"#I\"\"&\"$]\"\"\"'\"$/&\"\"(\"%_>\"\") \"%!H(\"\"*\"&gx#\"#5\"')z0\"\"#6\"'O`S\"#7\"(Qxb\"\"#8\"(qY+'\"#9\")? X?B\"#:\")S5()*)F)\"*!pZ([$\"#<\"+%=mcN\"\"#=\"+'GEzF&\"#>\",S?mw0#\"# ?-I\"OGI*protectedGFL6#\"\"\"\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 101 "dev_q_nonRootedFace_vertex_order2:=serie s(subs(gamma=dev_gamma,q_nonRootedFace_vertex_order2),z,N+1);" } {MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IBdev_q_nonRoo tedFace_vertex_order2G6\"+GI\"zGF%\"#;\"\"%\"#?\"\"&\"$W\"\"\"'\"$[%\" \"(\"%C=\"\")\"%/o\"\"*\"&+i#\"#5\"'k.5\"#6\"'WlQ\"#7\"(;:\\\"\"#8\"(c %pd\"#9\")!GhB#\"#:\")[g#o)F(\"*O'zwL\"#<\"+3_?:8\"#=\"+W/EH^\"#>\",?N $z-?F*-I\"OGI*protectedGFJ6#\"\"\"\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 87 "dev_q_vertex_vertex_order2:=ser ies(subs(gamma=dev_gamma,q_vertex_vertex_order2),z,N+1);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I;dev_q_vertex_vertex_ord er2G6\"+EI\"zGF%\"\"%F(\"#M\"\"'\"#%)\"\"(\"$'R\"\")\"%k9\"\"*\"%%z&\" #5\"&SC#\"#6\"&au)\"#7\"'W/M\"#8\"(mpK\"\"#9\"('*f<&\"#:\")w%3-#\"#;\" )3)o*y\"#<\"*#)[&)3$\"#=\"+sS'*37\"#>\"+Q//OZ\"#?-I\"OGI*protectedGFI6 #\"\"\"\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 69 "dev_qvv_orderNot2:=series(subs(gamma=dev_gamma,qvv_orderNot2), z,N+1);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I2de v_qvv_orderNot2G6\"+KI\"zGF%\"\"%\"\"#\"\"'\"\"$\"#GF(\"#!*\"\"&\"$M$F *\"%=7\"\"(\"%SX\"\")\"&Mq\"\"\"*\"&9W'\"#5\"'E\\C\"#6\"''oN*\"#7\"(!R )e$\"#8\")1n!Q\"\"#9\")Y:F`\"#:\"*30/1#\"#;\"*I.g)z\"#<\"+m\"465$\"#= \",m$[@17\"#>\",=:#o)p%\"#?-I\"OGI*protectedGFN6#\"\"\"\"#@" }{TEXT 20 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 45 "dev_q:=series (subs(gamma=dev_gamma,q),z,N+1);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I&dev_qG6\"+AI\"zGF%\"\"\"\"\"'\"\"%\"\")F)\"\"* \"#C\"#5\"#m\"#6\"$9#\"#7\"$w'\"#8\"%4A\"#9\"%'H(\"#:\"&gW#\"#;\"&EH) \"#<\"'oSG\"#=\"'#)=)*\"#>\"(=8U$\"#?-I\"OGI*protectedGFE6#F(\"#@" } {TEXT 20 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 229 "We perform th e sum of coefficients of the series of k-rooted 3-connected maps (whic h are in bijection via the Tutte's application with the k-rooted irred uctible quadrangulations) as indicated by the Burnside Lemma for plana r maps" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 312 "sum_k_rooted_3connected:=series(dev_q+z*subs(z=z**2,dev_q_rootedF ace_vertex_order2+dev_q_nonRootedFace_vertex_order2)+z**2*subs(z=z**2, dev_q_rootedFace_nonRootedFace_order2+dev_q_nonRootedFace_nonRootedFac e_order2)+subs(z=z**2,dev_q_vertex_vertex_order2)+sum(phi(k)*subs(z=z* *k,dev_qvv_orderNot2),k=3..N),z,N+1):" }{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 209 179 "We divide each coefficient of the pre ceding sum by 2*n to achieve the Burnside Lemma and obtain the first v alues of non-rooted 2-connected maps with respect to the number of edg es" }{TEXT 209 0 "" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 203 92 "n on_rooted_3connected_map:=series(1/2*int(series(sum_k_rooted_3connecte d/z,z,N+1),z),z,N+1);" }{MPLTEXT 1 203 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I:non_rooted_3connected_mapG6\"+AI\"zGF%\"\"\"\"\"'F( \"\")\"\"#\"\"*\"\"$\"#5\"\"%\"#6\"#:\"#7\"#K\"#8\"#*)\"#9\"$m#F1\"$(z \"#;\"%'\\#\"#<\"%7!)\"#=\"&Gg#\"#>\"&))e)\"#?-I\"OGI*protectedGFD6#F( \"#@" }{TEXT 20 0 "" }}}}}}{SECT 1 {PARA 268 "" 0 "" {TEXT 208 60 "Enu meration with respect to the number of vertices and faces" }{TEXT 208 0 "" }}{SECT 0 {PARA 274 "" 0 "" {TEXT 205 42 "Enumeration of unrooted unconstrained maps" }{TEXT 205 0 "" }}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 67 "Introduction of the algebraic series beta1(x1,x2) and be ta2(x1,x2) " }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 147 "We introduce the algebraic series beta1 and beta2 such that all serie s of k-rooted quadrangulations will be rational expressions of beta1 a nd beta2" }{TEXT 223 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 51 "Eq_x1_beta1_beta2:=x1-beta1+beta1**2+2*beta1*beta2;" } {MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I2Eq_x1_beta1_ beta2G6\",*I#x1GF%\"\"\"I&beta1GF%!\"\"*$F)\"\"#F(*&F)F(I&beta2GF%F(F, " }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 48 "Exp_ x1_beta1_beta2:=solve(Eq_x1_beta1_beta2,x1);" }{MPLTEXT 1 231 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I3Exp_x1_beta1_beta2G6\",(I&beta1GF% \"\"\"*$F'\"\"#!\"\"*&F'F(I&beta2GF%F(!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 51 "Eq_x2_beta1_beta2:=x2-beta2+bet a2**2+2*beta1*beta2;" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I2Eq_x2_beta1_beta2G6\",*I#x2GF%\"\"\"I&beta2GF%!\"\"* $F)\"\"#F(*&I&beta1GF%F(F)F(F," }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> \+ " 0 "" {MPLTEXT 1 231 48 "Exp_x2_beta1_beta2:=solve(Eq_x2_beta1_beta2, x2);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3Exp_x 2_beta1_beta2G6\",(I&beta2GF%\"\"\"*$F'\"\"#!\"\"*&I&beta1GF%F(F'F(!\" #" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 145 "We obtain iteratively the development of beta1 and beta2 in x1 and x2 (this wil l be used to find the first values of unrooted maps on the sphere)" } {TEXT 223 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 31 "n:=0:d ev_beta1:=0:dev_beta2:=0:" }{MPLTEXT 1 231 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 13 "while(n " 0 "" {MPLTEXT 1 231 222 "Eq1_diffBeta1x1_diffBet a2x1_beta1_beta2:=subs(diff(beta1(x1,x2),x1)=d_beta1_x1,diff(beta2(x1, x2),x1)=d_beta2_x1,beta1(x1,x2)=beta1,beta2(x1,x2)=beta2,diff(subs(bet a1=beta1(x1,x2),beta2=beta2(x1,x2),Eq_x1_beta1_beta2),x1));" } {MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IHEq1_diffBeta 1x1_diffBeta2x1_beta1_beta2G6\",,\"\"\"F'I+d_beta1_x1GF%!\"\"*&I&beta1 GF%F'F(F'\"\"#*&F(F'I&beta2GF%F'F,*&F+F'I+d_beta2_x1GF%F'F," }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 222 "Eq2_diffBeta 1x1_diffBeta2x1_beta1_beta2:=subs(diff(beta1(x1,x2),x1)=d_beta1_x1,dif f(beta2(x1,x2),x1)=d_beta2_x1,beta1(x1,x2)=beta1,beta2(x1,x2)=beta2,di ff(subs(beta1=beta1(x1,x2),beta2=beta2(x1,x2),Eq_x2_beta1_beta2),x1)); " }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IHEq2_diffB eta1x1_diffBeta2x1_beta1_beta2G6\",*I+d_beta2_x1GF%!\"\"*&I&beta2GF%\" \"\"F'F+\"\"#*&I+d_beta1_x1GF%F+F*F+F,*&I&beta1GF%F+F'F+F," }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 113 "solve(\{Eq1_dif fBeta1x1_diffBeta2x1_beta1_beta2,Eq2_diffBeta1x1_diffBeta2x1_beta1_bet a2\},\{d_beta1_x1,d_beta2_x1\});" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#<$/I+d_beta1_x1G6\",$*&,(!\"\"\"\"\"I&beta1GF&\"\" #I&beta2GF&F-F+,.*&F,F+F.F+\"\"%F+F+F,!\"%F.F2*$F,F-F1*$F.F-F1F*F*/I+d _beta2_x1GF&,$*&F.F+F/F*F-" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 88 "d_beta1_x1:=-(-1+2*beta1+2*beta2)/(4*beta1*beta2 +1-4*beta1-4*beta2+4*beta1^2+4*beta2^2);" }{MPLTEXT 1 231 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I+d_beta1_x1G6\",$*&,(!\"\"\"\"\"I&b eta1GF%\"\"#I&beta2GF%F,F*,.*&F+F*F-F*\"\"%F*F*F+!\"%F-F1*$F+F,F0*$F-F ,F0F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 74 "d_beta2_x1:=2*beta2/(4*beta1*beta2+1-4*beta1-4*beta2+4*beta1^2+4*b eta2^2);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+d _beta2_x1G6\",$*&I&beta2GF%\"\"\",.*&I&beta1GF%F)F(F)\"\"%F)F)F,!\"%F( F.*$F,\"\"#F-*$F(F0F-!\"\"F0" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 84 "We proceed similarly to find (beta1,beta2)-expression s for d_beta1_x2 and d_beta2_x2" }{TEXT 223 0 "" }}}{EXCHG {PARA 279 " > " 0 "" {MPLTEXT 1 231 222 "Eq1_diffBeta1x2_diffBeta2x2_beta1_beta2:= subs(diff(beta1(x1,x2),x2)=d_beta1_x2,diff(beta2(x1,x2),x2)=d_beta2_x2 ,beta1(x1,x2)=beta1,beta2(x1,x2)=beta2,diff(subs(beta1=beta1(x1,x2),be ta2=beta2(x1,x2),Eq_x1_beta1_beta2),x2));" }{MPLTEXT 1 231 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>IHEq1_diffBeta1x2_diffBeta2x2_beta1_ beta2G6\",*I+d_beta1_x2GF%!\"\"*&I&beta1GF%\"\"\"F'F+\"\"#*&F'F+I&beta 2GF%F+F,*&F*F+I+d_beta2_x2GF%F+F," }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 222 "Eq2_diffBeta1x2_diffBeta2x2_beta1_beta2: =subs(diff(beta1(x1,x2),x2)=d_beta1_x2,diff(beta2(x1,x2),x2)=d_beta2_x 2,beta1(x1,x2)=beta1,beta2(x1,x2)=beta2,diff(subs(beta1=beta1(x1,x2),b eta2=beta2(x1,x2),Eq_x2_beta1_beta2),x2));" }{MPLTEXT 1 231 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>IHEq2_diffBeta1x2_diffBeta2x2_beta1_ beta2G6\",,\"\"\"F'I+d_beta2_x2GF%!\"\"*&I&beta2GF%F'F(F'\"\"#*&I+d_be ta1_x2GF%F'F+F'F,*&I&beta1GF%F'F(F'F," }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 113 "solve(\{Eq1_diffBeta1x2_diffBeta2x2_ beta1_beta2,Eq2_diffBeta1x2_diffBeta2x2_beta1_beta2\},\{d_beta1_x2,d_b eta2_x2\});" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#< $/I+d_beta1_x2G6\",$*&I&beta1GF&\"\"\",.*&F)F*I&beta2GF&F*\"\"%F*F*F)! \"%F-F/*$F)\"\"#F.*$F-F1F.!\"\"F1/I+d_beta2_x2GF&,$*&,(F3F*F)F1F-F1F*F +F3F3" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 74 "d_beta1_x2:=2*beta1/(4*beta1*beta2+1-4*beta1-4*beta2+4*beta1^2+4*beta 2^2);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+d_be ta1_x2G6\",$*&I&beta1GF%\"\"\",.*&F(F)I&beta2GF%F)\"\"%F)F)F(!\"%F,F.* $F(\"\"#F-*$F,F0F-!\"\"F0" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 " " {MPLTEXT 1 231 88 "d_beta2_x2:=-(-1+2*beta1+2*beta2)/(4*beta1*beta2+ 1-4*beta1-4*beta2+4*beta1^2+4*beta2^2);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+d_beta2_x2G6\",$*&,(!\"\"\"\"\"I&beta1GF %\"\"#I&beta2GF%F,F*,.*&F+F*F-F*\"\"%F*F*F+!\"%F-F1*$F+F,F0*$F-F,F0F)F )" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 280 "" 0 "" {TEXT 232 84 "Obtainin g (beta1,beta2)-rational expressions for series of k-rooted quadrangul ations" }{TEXT 232 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 232 "As s eries of k-rooted quadrangulations can be written as derivatives of th e series F of k-rooted quadrangulations which is (beta1,beta2)-rationa l, we get (beta1,beta2)-rational expressions for the series of k-roote d quadrangulations" }{TEXT 223 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 59 "We use a well known (beta1,beta2)-rational expression for F" } {TEXT 223 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 83 "F:=fac tor(beta1*beta2*(1-2*beta1-2*beta2)/Exp_x1_beta1_beta2/Exp_x2_beta1_be ta2-1);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"F G6\",$*(,,I&beta1GF%!\"\"I&beta2GF%F**&F)\"\"\"F+F-\"\"&*$F)\"\"#F0*$F +F0F0F-,(F*F-F)F-F+F0F*,(F*F-F+F-F)F0F*F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 69 "We obtain (beta1,beta2)-rational expre ssions for the derivatives of F" }{TEXT 223 0 "" }}}{EXCHG {PARA 279 " > " 0 "" {MPLTEXT 1 231 66 "d_F_x1:=factor(diff(F,beta1)*d_beta1_x1+di ff(F,beta2)*d_beta2_x1);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I'd_F_x1G6\",$*&,(!\"\"\"\"\"I&beta1GF%F*I&beta2GF%\" \"#!\"#,(F)F*F,F*F+F-F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 " " {MPLTEXT 1 231 66 "d_F_x2:=factor(diff(F,beta1)*d_beta1_x2+diff(F,be ta2)*d_beta2_x2);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I'd_F_x2G6\",$*&,(!\"\"\"\"\"I&beta1GF%F*I&beta2GF%\"\"#F),(F)F *F,F*F+F-!\"#F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 79 "d_F_x1_x1:=factor(diff(d_F_x1,beta1)*d_beta1_x1+dif f(d_F_x1,beta2)*d_beta2_x1);" }{MPLTEXT 1 231 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 79 "d_F_x1_x2:=factor(diff(d_F_x1,beta1)* d_beta1_x2+diff(d_F_x1,beta2)*d_beta2_x2);" }{MPLTEXT 1 231 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I*d_F_x1_x2G6\",$**,(I&beta1GF%\"\"' !\"&\"\"\"I&beta2GF%F*F,,(!\"\"F,F)F,F-\"\"#!\"#,(F/F,F-F,F)F0F1,.*&F) F,F-F,\"\"%F,F,F)!\"%F-F6*$F)F0F5*$F-F0F5F/F/" }{TEXT 20 0 "" }}} {EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 79 "d_F_x2_x2:=factor(diff(d _F_x2,beta1)*d_beta1_x2+diff(d_F_x2,beta2)*d_beta2_x2);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I*d_F_x2_x2G6\",$**,&!\"# \"\"\"I&beta2GF%\"\"$F*,.*&I&beta1GF%F*F+F*\"\"%F*F*F/!\"%F+F1*$F/\"\" #F0*$F+F3F0!\"\",(F5F*F/F*F+F3F5,(F5F*F+F*F/F3!\"$F)" }{TEXT 20 0 "" } }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 189 "The series of k-rooted quadr angulations are expressible according to the derivatives of F, which a re (beta1,beta2)-rational. Hence we get (beta1,beta2)-rational express ions for these series" }{TEXT 223 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 157 "Ffb:=factor(2*Exp_x1_beta1_beta2**2*d_F_x1_x1+2*Ex p_x1_beta1_beta2*Exp_x2_beta1_beta2*d_F_x1_x2+5*Exp_x1_beta1_beta2*d_F _x1+2*Exp_x2_beta1_beta2*d_F_x2+F+1);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$FfbG6\"*(,&!\"\"\"\"\"I&beta2GF%\"\"#F), .*&I&beta1GF%F)F*F)\"\"%F)F)F.!\"%F*F0*$F.F+F/*$F*F+F/F(,(F(F)F*F)F.F+ F(" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 41 "Ff w:=subs(\{beta1=beta2,beta2=beta1\},Ffb);" }{MPLTEXT 1 231 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I$FfwG6\"*(,&!\"\"\"\"\"I&beta1GF%\" \"#F),.*&F*F)I&beta2GF%F)\"\"%F)F)F*!\"%F.F0*$F*F+F/*$F.F+F/F(,(F(F)F* F)F.F+F(" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 191 "Fff:=factor(2*Exp_x1_beta1_beta2**2*d_F_x1_x1+2*Exp_x2_beta1_beta 2**2*d_F_x2_x2+4*Exp_x1_beta1_beta2*Exp_x2_beta1_beta2*d_F_x1_x2+5*Exp _x1_beta1_beta2*d_F_x1+5*Exp_x2_beta1_beta2*d_F_x2+F+1);" }{MPLTEXT 1 231 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 91 "Fbb:=factor( 1/2*Exp_x1_beta1_beta2*Exp_x2_beta1_beta2*d_F_x1_x1+Exp_x2_beta1_beta2 *d_F_x1);" }{MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$ FbbG6\"*&I&beta2GF%\"\"\",.*&I&beta1GF%F(F'F(\"\"%F(F(F+!\"%F'F-*$F+\" \"#F,*$F'F/F,!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 41 "Fww:=subs(\{beta1=beta2,beta2=beta1\},Fbb);" } {MPLTEXT 1 231 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$FwwG6\"*&I&b eta1GF%\"\"\",.*&F'F(I&beta2GF%F(\"\"%F(F(F'!\"%F+F-*$F'\"\"#F,*$F+F/F ,!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 231 115 "Fbw:=factor(Exp_x1_beta1_beta2*Exp_x2_beta1_beta2*d_F_x1_x2+Exp_x 1_beta1_beta2*d_F_x1+Exp_x2_beta1_beta2*d_F_x2+F);" }{MPLTEXT 1 231 0 "" }}}}}{SECT 0 {PARA 281 "" 0 "" {TEXT 218 85 "Obtaining the initial \+ coefficients counting unrooted maps with i vertices and j faces" } {TEXT 218 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 245 "From Burnside Lemma, we will get the coefficients counting unrooted maps with i ver tices and j faces. We will obtain these coefficients for i+j <=N. Firs t we need to obtain the development in x1 and x2 of all series of k-r ooted quadrangulations" }{TEXT 206 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 213 "We get developments in x1 and x2 of F and of the series of k-rooted quadrangulations by replacing beta1 and beta2 by their de velopment in s1 and x2 in the (beta1,beta2)-rational expressions of ea ch of these series" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "dev_F:=expand(series(subs(beta1=dev_beta1,beta2=dev_b eta2,F),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "dev_Fff:=expand(series(subs(beta1=dev_beta1,beta2=dev _beta2,Fff),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "dev_Ffb:=expand(series(subs(beta1=dev_beta1,beta2=de v_beta2,Ffb),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 " " {MPLTEXT 1 0 73 "dev_Ffw:=expand(series(subs(beta1=dev_beta1,beta2=d ev_beta2,Ffw),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "dev_Fbb:=expand(series(subs(beta1=dev_beta1,beta2= dev_beta2,Fbb),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "dev_Fww:=expand(series(subs(beta1=dev_beta1,beta2 =dev_beta2,Fww),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "dev_Fbw:=expand(series(subs(beta1=dev_beta1,beta 2=dev_beta2,Fbw),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "dev_Fbw:=expand(series(subs(beta1=dev_beta1,bet a2=dev_beta2,Fbw),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 212 "We perform the sum of coefficients of the series o f k-rooted maps as indicated by Burnside Lemma for maps on the sphere \+ (Liskovets formula). This formula can be easily adapted to get an equa tion on series follows" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 402 "sum_k_rooted_maps_two_variables:=series(expand(conve rt(dev_F+x2*t*subs(x1=x1**2,x2=x2**2,t=t**2,dev_Ffb)+x1*t*subs(x1=x1** 2,x2=x2**2,t=t**2,dev_Ffw)+x1*x2*t**2*subs(x1=x1**2,x2=x2**2,t=t**2,de v_Fff)+x1/x2*sum(phi(k)*subs(x1=x1**k,x2=x2**k,t=t**k,dev_Fbb),k=2..N) +x2/x1*sum(phi(k)*subs(x1=x1**k,x2=x2**k,t=t**k,dev_Fww),k=2..N)+sum(p hi(k)*subs(x1=x1**k,x2=x2**k,t=t**k,dev_Fbw),k=2..N),polynom)),t,N+1): " }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 286 "We divide each coefficien t of the preceding sum by 2*(i+j) to achieve the Burnside Lemma and ob tain the first values of non-rooted planar maps with respect to the nu mber of vertices and faces. The coefficient of x1^i*x2^j is the number of unrooted maps having i+1 vertices and j+1 faces" }{TEXT 223 0 "" } }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 107 "non_rooted_maps_two_var iables:=series(1/2*int(convert(sum_k_rooted_maps_two_variables/t,polyn om),t),t,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6# >I>non_rooted_maps_two_variablesG6\"+KI\"tGF%,&I#x1GF%\"\"\"I#x2GF%F*F *,(*&F)F*F+F*\"\"#*$F+F.F**$F)F.F*F.,**$F+\"\"$F.*&F)F.F+F*\"\"&*$F)F3 F.*&F)F*F+F.F5F3,,*$F)\"\"%F3*&F)F.F+F.\"#B*&F)F*F+F3\"#9*$F+F:F3*&F)F 3F+F*F>F:,.*&F)F.F+F3\"$3\"*$F)F5\"\"'*$F+F5FE*&F)F3F+F.FC*&F)F*F+F:\" #U*&F)F:F+F*FIF5,0*$F+FEF>*&F)F5F+F*\"$S\"*&F)F*F+F5FN*&F)F3F+F3\"$h(* &F)F:F+F.\"$,&*$F)FEF>*&F)F.F+F:FSFE,2*$F)\"\"(\"#M*&F)F.F+F5\"%kA*&F) F:F+F3\"%WZ*&F)F*F+FE\"$t%*&F)FEF+F*Fin*&F)F3F+F:Fgn*&F)F5F+F.Fen*$F+F XFYFX,4*&F)F*F+FX\"%q;*&F)FEF+F.\"&(35*&F)F3F+F5\"&ox#*&F)F5F+F3Fdo*&F )FXF+F*F`o*$F+\"\")\"#&**$F)FhoFio*&F)F:F+F:\"&&\\Q*&F)F.F+FEFboFho,6* &F)F*F+Fho\"%pf*&F)FEF+F3\"'oO:*&F)F:F+F5\"')pz#*&F)F3F+FEFbp*&F)F.F+F X\"&5V%*&F)FXF+F.Fgp*$F)\"\"*\"$!G*$F+FjpF[q*&F)F5F+F:Fdp*&F)FhoF+F*F` pFjp,8*$F)\"#5\"$a)*$F+FaqFbq*&F)FhoF+F.\"'WC>*&F)FjpF+F*\"&z;#*&F)F:F +FE\"(QO(=*&F)FXF+F3\"'hm\")*&F)F3F+FXF[r*&F)F*F+FjpFgq*&F)F5F+F5\"(k# eC*&F)FEF+F:Fiq*&F)F.F+FhoFeqFaq,:*$F)\"#6\"%%p#*&F)FEF+F5\")K'4&>*&F) F*F+Faq\"&>%z*&F)F:F+FX\")kfz6*&F)F.F+Fjp\"'w\"G)*&F)FaqF+F*Fir*&F)Fjp F+F.F]s*&F)FhoF+F3\"(!)4?%*$F+FdrFer*&F)F5F+FEFgr*&F)FXF+F:F[s*&F)F3F+ FhoFasFdr,<*&F)FEF+FE\"*'[#\\!=*&F)F:F+Fho\")V)>2(*&F)F3F+Fjp\")Z81@*& F)FdrF+F*\"''\\$H*&F)FhoF+F:Fjs*&F)FXF+F5\"*;w:V\"*&F)FjpF+F3F\\t*$F+ \"#7\"%9()*&F)F.F+Faq\"(6t`$*$F)FdtFet*&F)F5F+FXFat*&F)FaqF+F.Fgt*&F)F *F+FdrF^tFdt,>*&F)FEF+FX\"+Gv2B:*&F)F*F+Fdt\"(154\"*&F)F5F+Fho\"*!oyo) **&F)FjpF+F:\"*g&pvS*&F)F:F+FjpFdu*&F)F3F+Faq\"*)GpL5*&F)FdrF+F.\")GV, :*$F+\"#8\"&S'G*&F)FdtF+F*F`u*&F)F.F+FdrFiu*&F)FaqF+F3Fgu*&F)FXF+FEF^u *$F)F[vF\\v*&F)FhoF+F5FbuF[v,@*&F)F5F+Fjp\"+W[Lmk*&F)FjpF+F5Fev*&F)F.F +Fdt\")csRj*&F)FaqF+F:\"+))GqtA*$F)F>\"&Sc**&F)F3F+Fdr\"*y`\\)\\*&F)Fd tF+F.Fhv*&F)FEF+Fho\",6$y$R>\"*&F)FXF+FX\",d(oah9*&F)FhoF+FEFaw*&F)F:F +FaqFjv*$F+F>F\\w*&F)F*F+F[v\"(8#yS*&F)F[vF+F*Fhw*&F)FdrF+F3F^wF>,B*&F )FjpF+FE\",/+]1\"))*&F)F:F+Fdr\",]C\")]@J:*&F)FdrF+F:F_x*&F)FaqF+F5Fdx*&F)F[vF+F.Fhx*$ F+\"#:\"''RB$*$F)F`yFay*&F)F3F+FdtFax*&F)F>F+F*F[yF`y,D*&F)FhoF+Fho\". hi%H%GF\"*$F+\"#;\"(N`5\"*&F)F`yF+F*\")I5sd*&F)F*F+F`yF\\z*&F)FjpF+FX \".*[K`yj5*&F)F[vF+F3\",9s%466*&F)F5F+Fdr\"-Zs([DY#*&F)FaqF+FE\"-$>T\" )>='*&F)F3F+F[vFaz*&F)F>F+F.\"+FL6;6*&F)FdtF+F:\",`?Csa'*&F)FEF+FaqFez *&F)FdrF+F5Fcz*&F)F:F+FdtFjz*$F)FiyFjy*&F)FXF+FjpF_z*&F)F.F+F>FhzFiy,F *&F)FEF+Fdr\".kcsxf:%*&F)FdrF+FEFc[l*&F)F>F+F3\",C5'Qb^*&F)FiyF+F*\"*K QL=#*&F)F5F+Fdt\"./.$Q?[9*&F)FhoF+Fjp\"/'pD&)yP;\"*&F)FXF+Faq\".KS!o6y #)*&F)F:F+F[v\"-sIIK/M*$F)\"#<\"()z8Q*&F)FjpF+FhoF\\\\l*&F)FaqF+FXF^ \\l*&F)FdtF+F5Fj[l*&F)F[vF+F:F`\\l*$F+Fb\\lFc\\l*&F)F.F+F`y\"+gT*)eY*& F)F*F+FiyFh[l*&F)F`yF+F.Fj\\l*&F)F3F+F>Ff[lFb\\l,H*&F)FiyF+F.\",=hy\"R >*&F)FdrF+FX\"/AnAUF+F:\".a20 <(R<*&F)F3F+F`y\"-$>V=#pB*&F)F:F+F>F^^l*&F)FdtF+FE\"/GK9&)=$p#*&F)F.F+ FiyF`]l*&F)FhoF+FaqF[^l*&F)FXF+FdrFb]l*&F)F5F+F[v\".M^N)*eH)*&F)F`yF+F 3F`^l*&F)FEF+FdtFc^l*&F)Fb\\lF+F*Fd]l*&F)F[vF+F5Fh^lFh]l,J*&F)F>F+F5\" /Gl+`VWY*&F)F`yF+F:\".CVjyjv)*$F)\"#>\")e$4l%*&F)FdrF+Fho\"0C_j(p00\") *&F)F[vF+FE\"0G\")G:![!p\"*&F)FEF+F[vFh_l*&F)FaqF+Fjp\"13!R:*>[-6*&F)F XF+Fdt\"0kAXYDcN%*$F+Fc_lFd_l*&F)FdtF+FXF]`l*&F)F.F+Fb\\l\",WdZ70)*&F) F:F+F`yFa_l*&F)F5F+F>F__l*&F)FhoF+FdrFf_l*&F)F*F+Fh]l\"+:'p<:$*&F)Fh]l F+F*Ff`l*&F)Fb\\lF+F.Fa`l*&F)F3F+Fiy\".[f*\\kz5*&F)FjpF+FaqF[`l*&F)Fiy F+F3Fj`lFc_l,L*&F)F>F+FE\"1Mx=%QW=.\"*&F)FhoF+Fdt\"1kd)))H\\rF'*&F)FEF +F>F_al*&F)F`yF+F5\"0nb`hf\"[D*&F)F[vF+FX\"1=CfPX?\")H*&F)F*F+Fc_l\",` 2(3-7*&F)F5F+F`yFdal*&F)F:F+Fiy\"/_6%=I%[V*&F)FaqF+Faq\"2'4`Z&e4.8\"*& F)FjpF+Fdr\"1U))[5QWj(**&F)FXF+F[vFfal*$F+\"#?\"*]w5k\"*$F)FbblFcbl*&F )F3F+Fb\\l\".rU6'>$)[*&F)FdtF+FhoFaal*&F)FdrF+FjpF_bl*&F)Fb\\lF+F3Ffbl *&F)F.F+Fh]l\"-=E!p`L$*&F)Fh]lF+F.F[cl*&F)Fc_lF+F*Fhal*&F)FiyF+F:F[blF bbl" }{TEXT 20 0 "" }}}}}{SECT 0 {PARA 274 "" 0 "" {TEXT 205 40 "Enume ration of unrooted 2-connected maps" }{TEXT 205 0 "" }}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 132 "All series of k-rooted simple quadrangulati ons composed with (x1,x2) -> (y1,y2)=(x1*(1+F)**2,x2*(1+F)**2) are (be ta1,beta2)-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 278 "In this section, from the equations linking series of k-roote d quadrangulations and series of k-rooted simple quadrangulations, we \+ get (beta1,beta2)-rational expressions for the series of k-rooted simp le quadrangulations where (y1,y2) is substituted by (x1*(1+F)**2,x2*(1 +F)**2)" }{TEXT 223 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 246 "We introduce a few series which will appear in the equations linking ser ies of k-rooted quadrangulations and series of k-rooted simple quadran gulations. d_f_x1 and d_f_x2 denote respectively the partial derivativ es of f according to x1 (resp. x2)" }{TEXT 223 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 19 "f:=factor(F/(1+F));" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"*&,,I&beta1GF%!\"\"I& beta2GF%F)*&F(\"\"\"F*F,\"\"&*$F(\"\"#F/*$F*F/F/F,,(F)F,F(F/F*F/F)" } {TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 66 "d_f_x1: =factor(diff(f,beta1)*d_beta1_x1+diff(f,beta2)*d_beta2_x1);" } {MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I'd_f_x1G6\",$ *&,(!\"\"\"\"\"I&beta2GF%F*I&beta1GF%\"\"#F*,(F)F*F,F-F+F-!\"#F)" } {TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 66 "d_f_x2: =factor(diff(f,beta1)*d_beta1_x2+diff(f,beta2)*d_beta2_x2);" } {MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I'd_f_x2G6\",$ *&,(!\"\"\"\"\"I&beta1GF%F*I&beta2GF%\"\"#F*,(F)F*F+F-F,F-!\"#F)" } {TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 84 "Fbw_typ e_a:=factor((2*Exp_x1_beta1_beta2*d_f_x1+2*Exp_x2_beta1_beta2*d_f_x2)* (1+F));" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+Fb w_type_aG6\",$*&,&I&beta1GF%\"\"\"I&beta2GF%F*F*,(!\"\"F*F)\"\"#F+F.F- !\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 71 "E: =factor(2*Exp_x1_beta1_beta2*d_F_x1+2*Exp_x2_beta1_beta2*d_F_x2+F+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"EG6\"*&,(! \"\"\"\"\"I&beta1GF%F)I&beta2GF%\"\"#F(,(F(F)F+F)F*F,F(" }{TEXT 20 0 " " }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 245 "We use the equations give n by the tree-decomposition by multiple edges and obtained in the arti cle to find (beta1,beta2)-rational expressions of series of k-rooted s imple quadrangulations where (y1,y2) is substituted by (x1*(1+F)**2,x2 *(1+F)**2)" }{TEXT 223 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 25 "gbb:=factor(Fbb*(1+F)/E);" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gbbG6\",$*(,(!\"\"\"\"\"I&beta1GF%\"\"#I&beta2 GF%F,F*F-F*,.*&F+F*F-F*\"\"%F*F*F+!\"%F-F1*$F+F,F0*$F-F,F0F)F)" } {TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 41 "gww:=su bs(\{beta1=beta2,beta2=beta1\},gbb);" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gwwG6\",$*(,(!\"\"\"\"\"I&beta1GF%\"\"#I &beta2GF%F,F*F+F*,.*&F+F*F-F*\"\"%F*F*F+!\"%F-F1*$F+F,F0*$F-F,F0F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 38 "gbw:=fac tor((Fbw-Fbw_type_a)*(F+1)/E);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gbwG6\",$*(I&beta1GF%\"\"\"I&beta2GF%F),.*&F(F)F*F) \"\"%F)F)F(!\"%F*F.*$F(\"\"#F-*$F*F0F-!\"\"F-" }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 19 "gfb:=factor(Ffb/E);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "gfw:= subs(\{beta1=beta2,beta2=beta1\},gfb);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gfwG6\"*(,&!\"\"\"\"\"I&beta1GF%\"\"#F), .*&F*F)I&beta2GF%F)\"\"%F)F)F*!\"%F.F0*$F*F+F/*$F.F+F/F(,(F(F)F.F)F*F+ F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 25 "gff: =factor(Fff/(1+F)/E);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gffG6\"*,,(!\"\"\"\"\"I&beta1GF%F)I&beta2GF%F)F),(F( F)F*F)F+\"\"#F),(F(F)F+F)F*F-F),.*&F*F)F+F)\"\"%F)F)F*!\"%F+F2*$F*F-F1 *$F+F-F1F(,(F(F)F*F-F+F-F(" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 66 "Introduction of the algebraic series eta1(y1,y2) and \+ eta2(y1,y2) " }{TEXT 204 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 285 "We make a change of variable by introducing the variable y1 and y 2 such that y1=x1*(1+F)**2 and y2=x2*(1+F)**2. These two variables wil l be the variables for the generating series of the k-rooted simple qu adrangulations. We calculate the (beta1,beta2)-rational expression of \+ y1 and y2 " }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 56 "Exp_y1_beta1_beta2:=factor(Exp_x1_beta1_beta2*(1+F)**2);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3Exp_y1_beta1_b eta2G6\",$**,(!\"\"\"\"\"I&beta1GF%\"\"#I&beta2GF%F,F,F+F*,(F)F*F+F*F- F,F),(F)F*F-F*F+F,!\"#F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 56 "Exp_y2_beta1_beta2:=factor(Exp_x2_beta1_beta2*(1+F)* *2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3Exp_y2_ beta1_beta2G6\",$**,(!\"\"\"\"\"I&beta1GF%\"\"#I&beta2GF%F,F,F-F*,(F)F *F+F*F-F,!\"#,(F)F*F-F*F+F,F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 234 165 "We introduce the algebraic series eta1 and eta2 s uch that all the generating series of k-rooted simple quadrangulations will be rational expressions of eta1 and eta2" }{TEXT 234 0 "" }}} {EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 38 "Eq_y1_eta1_eta2:=-y1+eta 1*(1-eta2)**2;" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 " 6#>I0Eq_y1_eta1_eta2G6\",&I#y1GF%!\"\"*&I%eta1GF%\"\"\",&F+F+I%eta2GF% F(\"\"#F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 38 "Eq_y2_eta1_eta2:=-y2+eta2*(1-eta1)**2;" }{MPLTEXT 1 233 0 "" } }{PARA 273 "" 1 "" {XPPMATH 20 "6#>I0Eq_y2_eta1_eta2G6\",&I#y2GF%!\"\" *&I%eta2GF%\"\"\",&F+F+I%eta1GF%F(\"\"#F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 35 "Exp_y1_eta1_eta2:=eta1*(1-eta2) **2;" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I1Exp_y 1_eta1_eta2G6\"*&I%eta1GF%\"\"\",&F(F(I%eta2GF%!\"\"\"\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 35 "Exp_y2_eta1_eta2 :=eta2*(1-eta1)**2;" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I1Exp_y2_eta1_eta2G6\"*&I%eta2GF%\"\"\",&F(F(I%eta1GF% !\"\"\"\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 234 79 "W e give the beginning of the development of eta1 and eta2 in terms of y 1 and y2" }{TEXT 234 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 24 "dev_eta1:=0:dev_eta2:=0:" }{MPLTEXT 1 233 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 20 "for n from 1 to N do" } {MPLTEXT 1 233 0 "" }{MPLTEXT 1 233 71 "\ndev_eta1:=factor(convert(ser ies(y1*t/(1-dev_eta2)**2,t,n+1),polynom)):" }{MPLTEXT 1 233 0 "" } {MPLTEXT 1 233 71 "\ndev_eta2:=factor(convert(series(y2*t/(1-dev_eta1) **2,t,n+1),polynom)):" }{MPLTEXT 1 233 0 "" }{MPLTEXT 1 233 4 "\nod:" }{MPLTEXT 1 233 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 234 461 "We hav e the intuition that beta1 and beta2 must be rational expressions of e ta1 and eta2 (if this is the case all generating series of simple quad rangulations, which are rational expressions of beta1 and beta2, will \+ be rational expressions of eta1 and eta2, which we expect). First, by \+ substituting y1 and y2 by their (beta1,beta2)-rational expressions in \+ eq_y1_eta1_eta2 and in eq_y2_eta1_eta2, we get two polynomial equation s linking beta1, beta2, eta1, eta2)" }{TEXT 234 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 86 "Eq1_eta1_eta2_beta1_beta2:=factor(num er(subs(y1=Exp_y1_beta1_beta2,Eq_y1_eta1_eta2)));" }{MPLTEXT 1 233 0 " " }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I:Eq1_eta1_eta2_beta1_beta2G6\", do*$I&beta1GF%\"\"$\"\"%F(\"\"\"*&F(F+I&beta2GF%F+!\"%*&I%eta1GF%F+I%e ta2GF%F+\"\"#*&F0F+F-F2!\"&*&F0F+F(F)F**&F0F+F-F)F2*&F0F+F(F+\"\"&*&F0 F+F-F+F**&F0F+F(F2!\")*&F0F+F1F2!\"\"*(F0F+F(F+F-F2\"\"**(F0F+F1F+F(F+ !#5*(F0F+F1F+F-F+F;*(F0F+F1F2F(F+F8*(F0F+F1F2F-F+F**$F(F2F.*&F(F2F-F+ \"\")*&F(F+F-F2F**(F0F+F(F+F-F+!#9*(F0F+F1F+F-F2\"#5*(F0F+F1F+F(F2\"#; *(F0F+F1F+F(F)F;*(F0F+F1F+F-F)F.*(F0F+F1F2F-F2F4*(F0F+F1F2F(F2F;*(F0F+ F1F2F(F)F**(F0F+F1F2F-F)F2*(F0F+F(F2F-F+\"#7F0F=**F0F+F1F+F(F+F-F+\"#G **F0F+F1F+F(F+F-F2!#=**F0F+F1F+F(F2F-F+!#C**F0F+F1F2F(F2F-F+FV**F0F+F1 F2F(F+F-F2F?**F0F+F1F2F(F+F-F+FJ" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 " > " 0 "" {MPLTEXT 1 233 86 "Eq2_eta1_eta2_beta1_beta2:=factor(numer(su bs(y2=Exp_y2_beta1_beta2,Eq_y2_eta1_eta2)));" }{MPLTEXT 1 233 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I:Eq2_eta1_eta2_beta1_beta2G6\",do*( I%eta2GF%\"\"\"I&beta1GF%F)I&beta2GF%F)!#9*$F+\"\"$\"\"%*(F(F)F*F)F+\" \"#\"#7*(F(F)F*F1F+F)\"\"**&F*F)F+F)!\"%F+F)*&I%eta1GF%F)F(F)F1*&F(F)F +F1!\")*&F(F)F*F1!\"&*&F(F)F*F.F1*&F(F)F+F.F/*(F(F)F8F1F+F.F/*(F(F)F8F 1F*F)F/*(F(F)F8F1F+F)\"\"&*(F(F)F8F1F*F1F<*(F(F)F8F1F+F1F:**F(F)F8F1F* F)F+F)F,**F(F)F8F1F*F)F+F1F2**F(F)F8F1F*F1F+F)F4*(F8F)F(F)F*F)F:*(F8F) F(F)F+F)!#5*&F*F1F+F)F/*&F*F)F+F1\"\")*&F(F)F*F)F/*&F(F)F+F)FB*&F(F)F8 F1!\"\"*(F(F)F8F1F*F.F1*(F8F)F(F)F+F1\"#;*(F8F)F(F)F*F1\"#5*(F8F)F(F)F *F.F6*(F8F)F(F)F+F.F:F(FQ*$F+F1F6**F8F)F(F)F*F)F+F)\"#G**F8F)F(F)F*F)F +F1!#C**F8F)F(F)F*F1F+F)!#=" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 97 "Eq_eta1_eta2_beta1:=factor(resultant(Eq1_eta1_e ta2_beta1_beta2,Eq2_eta1_eta2_beta1_beta2,beta2));" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3Eq_eta1_eta2_beta1G6\",$**I&b eta1GF%\"\"#,&!\"\"\"\"\"F(F)F),.*(I%eta1GF%F,I%eta2GF%F,F(F,\"\"$*&F0 F,F(F,F+F(F+*&F/F,F(F,F+F/F,*&F/F,F0F,F+F,,bv*&F/F)F(F1!#K*(F(\"\"%F0F 9F/\"\"&!#L*(F(F9F/F:F0F1\"#V*(F(F9F0F:F/F:\"\"**(F(F9F0F9F/F9\"$D\"*( F(F9F0F1F/F9!$j\"*(F(F9F/F9F0F:F;*(F(F9F/F1F0F:F=*(F(F9F/F9F0F)\"$6\"* (F(F1F0F9F/F:\"#`*(F(F1F/F:F0F1!#l*&F/F)F0F9F:*(F(F,F0F9F/F:F1*(F(F,F/ F:F0F1!\"$*(F(F,F0F:F/F:F+*(F(F,F0F9F/F9!#7*(F(F,F0F1F/F9\"#7*(F(F,F/F 9F0F:F9*(F(F,F/F1F0F:!\"'*(F(F,F/F9F0F)!\"%*(F(F,F/F1F0F9\"#=*(F(F,F/F )F0F9!#W*(F(F,F/F)F0F:\"#5*(F(F,F0F)F/F:F,*(F0F,F/F)F(F,\"#Q*(F0F,F/F) F(F)!#%**(F(F)F0F9F/F:!#B*(F(F)F/F:F0F1\"#D*(F(F)F0F:F/F:\"\"(*(F(F)F0 F9F/F9\"#\"**(F(F)F0F1F/F9!$,\"*(F(F)F/F9F0F:!#F*(F(F)F/F1F0F:F\\o*(F( F)F/F9F0F)\"#P*(F(F)F/F1F0F9!$G\"*(F(F)F/F)F0F9\"$]\"*(F(F)F/F)F0F:!#O *(F(F)F0F)F/F:!\"**(F(F1F0F:F/F:!#:*(F(F1F0F9F/F9!$/#*(F(F1F0F1F/F9\"$ c#*(F(F1F/F9F0F:\"#c*(F(F1F/F1F0F:!#v*(F(F1F/F9F0F)!$3\"*(F(F1F/F1F0F9 \"$t#*(F(F1F/F)F0F9!$A#*(F(F1F/F)F0F:\"#a*(F(F1F0F)F/F:\"#F*&F(F9F0F) \"#;*&F(F9F/F)F[r*(F(F9F/F)F0F:Fjo*(F(F9F0F)F/F:Fjo*(F(F9F/F1F0F9FC*(F (F9F/F)F0F9FG*&F/F)F0F:F+*&F/F)F0F)Fin*&F/F)F0F1!#5*&F0F,F/F)!\"&*(F0F ,F/F)F(F1\"#!)*(F0F)F/F1F(F,\"\"'*(F0F1F/F1F(F,!#=*(F/F)F0F)F(F,!#w*(F /F)F0F1F(F,Fhr*(F0F1F(F)F/F,!\")*(F/F,F0F,F(F)Fas*(F/F,F0F,F(F1\"#K*(F /F,F0F)F(F)F[r*(F/F,F0F)F(F1!#c*(F/F)F0F)F(F)\"$%>*(F/F)F0F1F(F)!$Q#*( F0F,F/F1F(F)FW*(F0F)F/F1F(F)!#S*(F0F1F/F1F(F)\"$O\"*(F0F,F/F1F(F9!#C*( F0F,F/F)F(F9Fas*(F0F)F/F1F(F9!$P\"*(F0F1F/F1F(F9\"$$>*(F/F,F0F,F(F9F7* (F/F)F0F)F(F9\"$\\\"*(F/F)F0F1F(F9Fet*(F0F)F/F1F(F1\"$.\"*(F0F1F/F1F(F 1!$D$*(F/F)F0F)F(F1!$-#*(F/F)F0F1F(F1\"$A$*(F0F,F/F1F(F1\"#C*(F0F)F(F9 F/F,Fas*(F0F1F(F9F/F,Fbt*(F0F1F(F1F/F,Feu*$F/F)F,*&F/F)F(F,Fas*&F/F)F( F)FeuF,F9" }{TEXT 20 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 234 163 "W e conjecture that the factor which is of degree one in beta1 is alway s null, and thus conjecture from it a rational expression of beta1 in \+ terms of eta1 and eta2" }{TEXT 234 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 95 "Exp_beta1_eta1_eta2:=solve(3*eta1*eta2*beta1-eta1*b eta1-beta1-eta2*beta1+eta1-eta1*eta2,beta1);" }{MPLTEXT 1 233 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I4Exp_beta1_eta1_eta2G6\"*(I%eta1GF% \"\"\",&!\"\"F(I%eta2GF%F(F(,**&F'F(F+F(\"\"$F+F*F*F(F'F*F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 234 178 "It is logical by a \+ clear argument of symmetry to conjecture that the rational expression \+ of beta2 in terms of eta1 and eta2 is the expression of beta1 by excha nging eta1 and eta2" }{TEXT 234 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 69 "Exp_beta2_eta1_eta2:=subs(\{eta1=eta2,eta2=eta1\},E xp_beta1_eta1_eta2);" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I4Exp_beta2_eta1_eta2G6\"*(I%eta2GF%\"\"\",&!\"\"F(I%e ta1GF%F(F(,**&F+F(F'F(\"\"$F'F*F*F(F+F*F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 234 275 "We check that these two rational expr essions are correct by checking that the two following expressions are null (in fact we check that the relations y1=x1*(1+F)**2 and y2=x2*(1 +F)**2 remain true when we replace y1,y2,x1,x2 and F by their expressi on in terms of eta1 and eta2" }{TEXT 234 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 154 "factor(subs(beta1=eta1*(-1+eta2)/(3*eta1*eta 2-eta1-1-eta2),beta2=eta2*(-1+eta1)/(-1-eta2+3*eta1*eta2-eta1),Exp_x1_ beta1_beta2*(1+F)**2)-Exp_y1_eta1_eta2);" }{MPLTEXT 1 233 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 154 "factor(subs(beta1=eta1*(-1+eta 2)/(3*eta1*eta2-eta1-1-eta2),beta2=eta2*(-1+eta1)/(-1-eta2+3*eta1*eta2 -eta1),Exp_x2_beta1_beta2*(1+F)**2)-Exp_y2_eta1_eta2);" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 20 0 "" }}}} {SECT 0 {PARA 275 "" 0 "" {TEXT 204 71 "All series of k-rooted simple \+ quadrangulations are (eta1,eta2)-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 181 "We just have to substitute beta1 and \+ beta2 by their expression according to eta1 and eta2 in each (beta1,be ta2)-rational expression of the series of k-rooted simple quadrangulat ions" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 75 "g ff:=factor(subs(beta1=Exp_beta1_eta1_eta2,beta2=Exp_beta2_eta1_eta2,gf f));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gffG6\" **,&*&I%eta1GF%\"\"\"I%eta2GF%F*F*!\"\"F*F*,&F,F*F+F*F,,&F,F*F)F*F,,*F (\"\"$F+F*F,F*F)F*F," }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 75 "gfb:=factor(subs(beta1=Exp_beta1_eta1_eta2,beta2=Exp_ beta2_eta1_eta2,gfb));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gfbG6\",$**,&\"\"\"F)I%eta1GF%F)F),&!\"\"F)I%eta2GF% F)F),&F,F)F*F)F,,**&F*F)F-F)\"\"$F-F)F,F)F*F)F,F," }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 75 "gfw:=factor(subs(beta1=Exp _beta1_eta1_eta2,beta2=Exp_beta2_eta1_eta2,gfw));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gfwG6\",$**,&!\"\"\"\"\"I%eta1GF %F*F*,&I%eta2GF%F*F*F*F*,&F)F*F-F*F),**&F+F*F-F*\"\"$F-F*F)F*F+F*F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 75 "gbb:= factor(subs(beta1=Exp_beta1_eta1_eta2,beta2=Exp_beta2_eta1_eta2,gbb)); " }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gbbG6\"*( I%eta2GF%\"\"\",&!\"\"F(I%eta1GF%F(F(,**&F+F(F'F(\"\"$F'F(F*F(F+F(F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 233 75 "gww:=f actor(subs(beta1=Exp_beta1_eta1_eta2,beta2=Exp_beta2_eta1_eta2,gww));" }{MPLTEXT 1 233 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gwwG6\"*(I %eta1GF%\"\"\",&!\"\"F(I%eta2GF%F(F(,**&F'F(F+F(\"\"$F+F(F*F(F'F(F*" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 75 "gbw:=fact or(subs(beta1=Exp_beta1_eta1_eta2,beta2=Exp_beta2_eta1_eta2,gbw));" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I$gbwG6\",$*(I%e ta1GF%\"\"\"I%eta2GF%F),**&F(F)F*F)\"\"$F*F)!\"\"F)F(F)F.!\"%" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 99 "We also obtain from \+ the article of Mullin and Schellenberg an (eta1,eta2)-rational express ion for g" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "g:=factor(eta1*eta2*(1-eta1-eta2)+Exp_y1_eta1_eta2+Exp_y2_eta1_eta 2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"gG6\",( *&I%eta1GF%\"\"\"I%eta2GF%F)!\"$F(F)F*F)" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 281 "" 0 "" {TEXT 218 97 "Obtaining the initial coefficients cou nting unrooted 2-connected maps with i vertices and j faces" }{TEXT 218 0 "" }}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 88 "Obtaining the develo pment in y1 and y2 of the series of k-rooted simple quadrangulations" }{TEXT -1 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 246 "From the rati onal expressions of the k-rooted simple quadrangulations in terms of e ta1 and eta2 and from the developments of eta1 and eta2 in terms of y1 and y2, we obtain developments of the k-rooted simple quadrangulation s in terms of y1 and y2" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "dev_gbb:=expand(series(subs(eta1=dev_eta1,eta2=dev_e ta2,gbb),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "dev_gww:=expand(series(subs(eta1=dev_eta1,eta2=dev_et a2,gww),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "dev_gbw:=expand(series(subs(eta1=dev_eta1,eta2=dev_et a2,gbw),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "dev_gfb:=expand(series(subs(eta1=dev_eta1,eta2=dev_et a2,gfb),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "dev_gfw:=expand(series(subs(eta1=dev_eta1,eta2=dev_et a2,gfw),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 69 "dev_gff:=expand(series(subs(eta1=dev_eta1,eta2=dev_et a2,gff),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 65 "dev_g:=expand(series(subs(eta1=dev_eta1,eta2=dev_eta2 ,g),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 43 "Apply Burnside Lemma to get the enumeration" }{TEXT -1 0 "" }} {EXCHG {PARA 270 "" 0 "" {TEXT 206 445 "We perform a sum of the develo ments of the series of the generating functions of the k-rooted simple quadrangulations in which we substitute y1=y1**k and y2=y2**k when th e generating serie corresponds to k-rooted simple quadrangulations of \+ order k. This sum corresponds fast to the Burnside formula to obtain n on rooted maps in terms of k-rooted maps of the same family of maps. I t just remains to divide the sum by two times the number of edges" } {TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 411 "sum_k_r ooted_non_separable_two_variables:=convert(expand(series(dev_g+y2*t*su bs(y1=y1**2,y2=y2**2,t=t**2,dev_gfb)+y1*t*subs(y1=y1**2,y2=y2**2,t=t** 2,dev_gfw)+y1*y2*t**2*subs(y1=y1**2,y2=y2**2,t=t**2,dev_gff)+y1/y2*sum (phi(k)*subs(y1=y1**k,y2=y2**k,t=t**k,dev_gbb),k=2..N)+y2/y1*sum(phi(k )*subs(y1=y1**k,y2=y2**k,t=t**k,dev_gww),k=2..N)+sum(phi(k)*subs(y1=y1 **k,y2=y2**k,t=t**k,dev_gbw),k=2..N),t,N+1)),polynom):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 202 "We achieve the Burnsid e formula by dividing the preceding sum by two times the number of edg es. The coefficient for y1^i*y2^j is the number of unrooted 2-connecte d maps having i+1 vertices and j+1 faces" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 112 "non_rooted_non_separable_map_two _variables:=series(1/2*int(sum_k_rooted_non_separable_two_variables/t, t),t,N+1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IKn on_rooted_non_separable_map_two_variablesG6\"+KI\"tGF%,&I#y1GF%\"\"\"I #y2GF%F*F**&F)F*F+F*\"\"#,&*&F)F-F+F*F**&F)F*F+F-F*\"\"$,(*&F)F*F+F1F* *&F)F-F+F-F**&F)F1F+F*F*\"\"%,**&F)F-F+F1F-*&F)F1F+F-F-*&F)F*F+F6F**&F )F6F+F*F*\"\"&,,*&F)F-F+F6F1*&F)F*&F)FiqF +F1\"%s7*&F)F1F+FiqFdt*&F)F*F+FcrF**&F)F*&F)F6F+FjpFbu*&F)F^sF+F-Fdu\"#:,@* &F)FguF+F*F**&F)F6F+Fiq\"&vS&*&F)FP*&F)FA F+FA\")&>0U\"*&F)FMF+F]o\")\\X>5*&F)F]oF+FMFcv*&F)F*F+FguF**&F)F[pF+FD F_v*&F)FjpF+F0\"*&F)FjpF+F]oF`\\l*&F)FguF+F\"*&F)FHF+F6Fb]l*&F+F*F)FduF**&F+FduF)F*F*\"#?" }{TEXT 20 0 "" }}} }}}{SECT 0 {PARA 274 "" 0 "" {TEXT 205 40 "Enumeration of unrooted 3-c onnected maps" }{TEXT 205 0 "" }}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 150 "Obtaining (eta1-eta2)-rational expressions for series linked to s imple quadrangulations and appearing in the equations given by the tre e-decomposition" }{TEXT 204 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 549 "First we need to find an (eta1,eta2)-rational expression for each partial derivative of eta1 and eta2. For example, to obtain an (eta1, eta2)-rational expression for d eta1/dy1 (noted d_eta1_y1) and d eta2/ dy1 (noted d_eta2_y1), we derivate respectively Eq_y1_eta1_eta2 and Eq _y2_eta1_eta2 according to y1. We thus obtain two linear equations lin king d_eta1_y1, d_eta2_y1, eta1 and eta2 and whose two unknown series \+ are d_eta1_y1 and d_eta2_y1. Solving this system of two equations, we \+ get (eta1,eta2)-rational expressions for d_eta1_y1 and d_eta2_y1" } {TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 204 "Eq1_dif fEta1y1_diffEta2y1_eta1_eta2:=subs(diff(eta1(y1,y2),y1)=d_eta1_y1,diff (eta2(y1,y2),y1)=d_eta2_y1,eta1(y1,y2)=eta1,eta2(y1,y2)=eta2,diff(subs (eta1=eta1(y1,y2),eta2=eta2(y1,y2),Eq_y1_eta1_eta2),y1));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IDEq1_diffEta1y1_diffEta2 y1_eta1_eta2G6\",(!\"\"\"\"\"*&I*d_eta1_y1GF%F(,&F(F(I%eta2GF%F'\"\"#F (*(I%eta1GF%F(F+F(I*d_eta2_y1GF%F(!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 204 "Eq2_diffEta1y1_diffEta2y1_eta1_e ta2:=subs(diff(eta1(y1,y2),y1)=d_eta1_y1,diff(eta2(y1,y2),y1)=d_eta2_y 1,eta1(y1,y2)=eta1,eta2(y1,y2)=eta2,diff(subs(eta1=eta1(y1,y2),eta2=et a2(y1,y2),Eq_y2_eta1_eta2),y1));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IDEq2_diffEta1y1_diffEta2y1_eta1_eta2G6\",&*&I*d_et a2_y1GF%\"\"\",&F)F)I%eta1GF%!\"\"\"\"#F)*(I%eta2GF%F)F*F)I*d_eta1_y1G F%F)!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 103 "solve(\{Eq1_diffEta1y1_diffEta2y1_eta1_eta2,Eq2_diffEta1y1_diffEt a2y1_eta1_eta2\},\{d_eta1_y1,d_eta2_y1\});" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#<$/I*d_eta2_y1G6\",$*&,.*&I%eta1GF&\" \"\"I%eta2GF&F,!\"#*&F+F,F-\"\"#\"\"$F,F,F+!\"\"F-F.*$F-F0F,F2F-F,F0/I *d_eta1_y1GF&,$*&,&F2F,F+F,F,F)F2F2" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 72 "d_eta1_y1:=-(-1+eta1)/(-eta1+1-2*eta2-2 *eta1*eta2+eta2^2+3*eta1*eta2^2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I*d_eta1_y1G6\",$*&,&!\"\"\"\"\"I%eta1GF%F*F*,.*& F+F*I%eta2GF%F*!\"#*&F+F*F.\"\"#\"\"$F*F*F+F)F.F/*$F.F1F*F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 68 "d_eta2_y1:=2*et a2/(-eta1+1-2*eta2-2*eta1*eta2+eta2^2+3*eta1*eta2^2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I*d_eta2_y1G6\",$*&,.*&I%eta1 GF%\"\"\"I%eta2GF%F+!\"#*&F*F+F,\"\"#\"\"$F+F+F*!\"\"F,F-*$F,F/F+F1F,F +F/" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 204 "Eq 1_diffEta1y2_diffEta2y2_eta1_eta2:=subs(diff(eta1(y1,y2),y2)=d_eta1_y2 ,diff(eta2(y1,y2),y2)=d_eta2_y2,eta1(y1,y2)=eta1,eta2(y1,y2)=eta2,diff (subs(eta1=eta1(y1,y2),eta2=eta2(y1,y2),Eq_y1_eta1_eta2),y2));" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IDEq1_diffEta1y2 _diffEta2y2_eta1_eta2G6\",&*&I*d_eta1_y2GF%\"\"\",&F)F)I%eta2GF%!\"\" \"\"#F)*(I%eta1GF%F)F*F)I*d_eta2_y2GF%F)!\"#" }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 204 "Eq2_diffEta1y2_diffEta2y2 _eta1_eta2:=subs(diff(eta1(y1,y2),y2)=d_eta1_y2,diff(eta2(y1,y2),y2)=d _eta2_y2,eta1(y1,y2)=eta1,eta2(y1,y2)=eta2,diff(subs(eta1=eta1(y1,y2), eta2=eta2(y1,y2),Eq_y2_eta1_eta2),y2));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 103 "solve(\{Eq1_diffEta1y2_diffEta2y 2_eta1_eta2,Eq2_diffEta1y2_diffEta2y2_eta1_eta2\},\{d_eta1_y2,d_eta2_y 2\});" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#<$/I*d_et a1_y2G6\",$*&,.*&I%eta1GF&\"\"\"I%eta2GF&F,!\"#*&F-F,F+\"\"#\"\"$F,F,F -!\"\"F+F.*$F+F0F,F2F+F,F0/I*d_eta2_y2GF&,$*&,&F2F,F-F,F,F)F2F2" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 68 "d_eta1_y2 :=2*eta1/(-eta2+1-2*eta1-2*eta1*eta2+eta1^2+3*eta2*eta1^2);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I*d_eta1_y2G6\", $*&,.*&I%eta1GF%\"\"\"I%eta2GF%F+!\"#*&F,F+F*\"\"#\"\"$F+F+F,!\"\"F*F- *$F*F/F+F1F*F+F/" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 72 "d_eta2_y2:=-(-1+eta2)/(-eta2+1-2*eta1-2*eta1*eta2+eta 1^2+3*eta2*eta1^2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 146 "We introduce a few useful generating functions in view \+ of the enumeration of 3-connected planar maps according to the number \+ of vertices and faces" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 47 "W:=factor(g-Exp_y1_eta1_eta2-Exp_y2_eta1_eta2);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"WG6\",$*(I%et a2GF%\"\"\"I%eta1GF%F),(F*F)!\"\"F)F(F)F)F," }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 36 "G:=factor(W/(1+W/Exp_y2_eta1_eta2 ));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"GG6\"*, I%eta2GF%\"\"\"I%eta1GF%F(,(F)F(!\"\"F(F'F(F(,&F+F(F)F(\"\"#,(F+F(F)F( *&F)F(F'F(F(F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 143 "B:=factor(Exp_y1_eta1_eta2*(diff(G,eta1)*d_eta1_y1+diff(G,eta 2)*d_eta2_y1)+Exp_y2_eta1_eta2*(diff(G,eta1)*d_eta1_y2+diff(G,eta2)*d_ eta2_y2)-G);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I \"BG6\"**,&!\"\"\"\"\"I%eta1GF%F)\"\"%F*F)I%eta2GF%F),(F(F)F*F)*&F*F)F ,F)F)!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 143 "C:=factor(Exp_y1_eta1_eta2*(diff(W,eta1)*d_eta1_y1+diff(W,eta2)*d _eta2_y1)+Exp_y2_eta1_eta2*(diff(W,eta1)*d_eta1_y2+diff(W,eta2)*d_eta2 _y2)-W);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"CG 6\"*&I%eta1GF%\"\"\"I%eta2GF%F(" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 237 "To calculate generating functions of k-rooted irr educible quadrangulations, we will also need the (eta1,eta2)-rational \+ expressions of the generating functions of the families of 2-rooted si mple quadrangulations introduced in the article " }{TEXT 206 0 "" }}} {EXCHG {PARA 270 "" 0 "" {TEXT 206 90 "As explained in the article, we guess the rational expression of these functions and guess" }{TEXT 206 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 65 "We guess a (eta1,et a2)-rational expression for g_nonRooted_rooted" }{TEXT 206 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 42 "g_nonRooted_rooted:=1/((1- eta1)*(1-eta2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 " 6#>I3g_nonRooted_rootedG6\"*&,&\"\"\"F(I%eta1GF%!\"\"F*,&F(F(I%eta2GF% F*F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 40 "We chec k that this expression is correct" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 259 "factor(Exp_y1_eta1_eta2*diff(g_nonRoot ed_rooted,eta1)*d_eta1_y1+Exp_y1_eta1_eta2*diff(g_nonRooted_rooted,eta 2)*d_eta2_y1+Exp_y2_eta1_eta2*diff(g_nonRooted_rooted,eta1)*d_eta1_y2+ Exp_y2_eta1_eta2*diff(g_nonRooted_rooted,eta2)*d_eta2_y2+g_nonRooted_r ooted-gff);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\" \"!" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 54 "g_n onRooted_nonRooted:=factor(gff-g_nonRooted_rooted);" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 68 "We guess an (eta1,eta2)-ra tional expression for g_rooted_blackVertex" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 36 "g_rooted_blackVertex:=eta1/(1-eta 1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I5g_rooted _blackVertexG6\"*&I%eta1GF%\"\"\",&F(F(F'!\"\"F*" }{TEXT 20 0 "" }}} {EXCHG {PARA 270 "" 0 "" {TEXT 206 40 "We check that this expression i s correct" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 281 "factor(2*Exp_y1_eta1_eta2*diff(g_rooted_blackVertex,eta1)*d_eta1_ y1+2*Exp_y1_eta1_eta2*diff(g_rooted_blackVertex,eta2)*d_eta2_y1+2*Exp_ y2_eta1_eta2*diff(g_rooted_blackVertex,eta1)*d_eta1_y2+2*Exp_y2_eta1_e ta2*diff(g_rooted_blackVertex,eta2)*d_eta2_y2+g_rooted_blackVertex-(gf b-1));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\"\"!" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 62 "g_nonRoot ed_blackVertex:=factor((gfb-1)-g_rooted_blackVertex);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 71 "g_rooted_whiteVert ex:=subs(\{eta1=eta2,eta2=eta1\},g_rooted_blackVertex);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I5g_rooted_whiteVertexG6\"* &I%eta2GF%\"\"\",&F(F(F'!\"\"F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 77 "g_nonRooted_whiteVertex:=subs(\{eta1=eta2,eta 2=eta1\},g_nonRooted_blackVertex);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I8g_nonRooted_whiteVertexG6\",$**I%eta2GF%\"\"\", &F)F)I%eta1GF%F)F),&!\"\"F)F(F)F-,**&F+F)F(F)\"\"$F(F)F-F)F+F)F-\"\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 394 "We also int roduce the generating functions l and h of 2-rooted simple quadrangula tions of type face-face (resp. face-vertex) such that the 2-root is in cident to an axial face. We distinguish h_blackVertex and h_whiteVerte x whether the axial vertex is black or white. We denote by tl the seri es l where the variables y1 and y2 are exchanged and define similarly \+ th_blackVertex and th_whiteVertex" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 53 "l:=factor(g_nonRooted_rooted/(1+W/Exp_y 1_eta1_eta2));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 34 "tl:=subs(\{eta1=eta2,eta2=eta1\},l);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I#tlG6\",$*(,&!\"\"\"\"\"I%et a1GF%F*F*,(F)F*F+F**&F+F*I%eta2GF%F*F*F),&F)F*F.F*F)F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 67 "h_blackVertex:=factor( g_rooted_blackVertex/(1+W/Exp_y1_eta1_eta2));" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I.h_blackVertexG6\"**I%eta1GF%\"\"\" ,&!\"\"F(I%eta2GF%F(\"\"#,(F*F(F+F(*&F'F(F+F(F(F*,&F*F(F'F(F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 58 "th_blackVertex: =subs(\{eta1=eta2,eta2=eta1\},h_blackVertex);" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I/th_blackVertexG6\"**I%eta2GF%\"\" \",&!\"\"F(I%eta1GF%F(\"\"#,(F*F(F+F(*&F+F(F'F(F(F*,&F*F(F'F(F*" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 88 "h_whiteVe rtex:=factor((g_rooted_whiteVertex-W/Exp_y1_eta1_eta2)/(1+W/Exp_y1_eta 1_eta2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.h_ whiteVertexG6\",$*(I%eta1GF%\"\"\"I%eta2GF%F),(!\"\"F)F*F)*&F(F)F*F)F) F,F," }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 58 "th _whiteVertex:=subs(\{eta1=eta2,eta2=eta1\},h_whiteVertex);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I/th_whiteVertexG6\",$*(I %eta2GF%\"\"\"I%eta1GF%F),(!\"\"F)F*F)*&F*F)F(F)F)F,F," }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 120 "All series of k-rooted i rreducible quadrangulations composed with (y1,y2)->(z1,z2)=(W/y2,W/y1 ) are (eta1,eta2)-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 212 "We fin d (eta1,eta2)-rational expressions for the generating functions of k-r ooted irreducible quadrangulations from the equations given by the tre e-decomposition by separating 4-cycles and obtained in the article" } {TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 75 "q_nonRoo ted_rooted:=factor((l-1-tl*W/Exp_y2_eta1_eta2)/g_nonRooted_rooted);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3q_nonRooted_ro otedG6\",$*,I%eta1GF%\"\"\"I%eta2GF%F),,!\"\"F)F*F)*&F(\"\"#F*F.F)*&F( F)F*F)F,F(F)F),(F,F)F(F)F/F)F,,(F,F)F*F)F/F)F,F," }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 137 "q_rooted_blackVertex:=fac tor(h_blackVertex-W/Exp_y2_eta1_eta2-th_whiteVertex*W/Exp_y2_eta1_eta2 -q_nonRooted_rooted*g_rooted_blackVertex);" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I5q_rooted_blackVertexG6\",$*(I%eta1 GF%\"\"#I%eta2GF%F),(!\"\"\"\"\"F(F-*&F(F-F*F-F-F,F," }{TEXT 20 0 "" } }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 71 "q_rooted_whiteVertex:=su bs(\{eta1=eta2,eta2=eta1\},q_rooted_blackVertex);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I5q_rooted_whiteVertexG6\",$*(I%et a2GF%\"\"#I%eta1GF%F),(!\"\"\"\"\"F(F-*&F*F-F(F-F-F,F," }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 132 "q_nonRooted_nonRooted :=factor((g_nonRooted_nonRooted-tl**2*C/Exp_y2_eta1_eta2-l**2*C/Exp_y1 _eta1_eta2)*W/(C*g_nonRooted_rooted**2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I6q_nonRooted_nonRootedG6\"*0,(I%eta1GF%\" \"\"!\"\"F)I%eta2GF%F)F)F(F)F+F),R*&F+F)F(\"\"$!\"$*&F+\"\"#F(F.\"#5*& F+F1F(\"\"%F.*&F(F1F+F.F2*&F+F.F(F.\"\")*&F+F.F(F4!#5F1F)*&F(F)F+F.F/* &F(F.F+F4F9*&F(F1F+F4F.F+!\"%*&F+F.F(\"\"&F**&F+F4F(F4F=*&F+F4F(F?F.*& F+F?F(F.F**&F+F?F(F4F.*&F+F?F(F?F1*$F(F1F1*&F+F)F(F1F)*&F(F1F+F1!#;*&F (F)F+F)\"\"'*&F(F)F+F1F)*$F+F1F1F(F=F),(F*F)F+F)FIF)!\"#,(F*F)F(F)FIF) FN,*FIF.F+F)F*F)F(F)F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 252 "q_nonRooted_blackVertex:=factor((g_nonRooted_blackVe rtex-2*C/Exp_y2_eta1_eta2*th_whiteVertex*tl-2*C/Exp_y1_eta1_eta2*h_bla ckVertex*l-2*C/Exp_y2_eta1_eta2*tl-2*C/W*q_nonRooted_nonRooted*g_nonRo oted_rooted*g_rooted_blackVertex)*W/(C*g_nonRooted_rooted));" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 77 "q_non Rooted_whiteVertex:=subs(\{eta1=eta2,eta2=eta1\},q_nonRooted_blackVert ex);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I8q_nonRo oted_whiteVertexG6\",$*.,(I%eta1GF%\"\"\"!\"\"F*I%eta2GF%F*F*,.*&F)\" \"#F,F/F/*&F)F*F,F/\"\"$*&F)F*F,F*!\"&F,!\"$*$F,F/F+\"\"%F*F*F,F/F)F/, (F+F*F,F*F2F*!\"#,*F2F1F,F*F+F*F)F*F+F/" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 367 "qww_order2:=factor((gww-B/Exp_y2 _eta1_eta2/(1-G/Exp_y2_eta1_eta2)-th_blackVertex**2*C/Exp_y2_eta1_eta2 **2*Exp_y1_eta1_eta2-h_whiteVertex**2*C/Exp_y2_eta1_eta2-2*h_whiteVert ex*C/Exp_y2_eta1_eta2-C/W*Exp_y1_eta1_eta2/Exp_y2_eta1_eta2*q_nonRoote d_nonRooted*g_rooted_whiteVertex**2-C/W*Exp_y1_eta1_eta2/Exp_y2_eta1_e ta2*q_nonRooted_whiteVertex*g_rooted_whiteVertex)*W/C);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 51 "qbb_order2:=subs (\{eta1=eta2,eta2=eta1\},qww_order2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+qbb_order2G6\"*2,(I%eta1GF%\"\"\"!\"\"F)I%et a2GF%F)F),>*&F(F)F+\"\"#!\"#*&F+F.F(\"\"%F.*&F+F.F(\"\"$F**&F(F.F+F.! \"$*&F+F)F(F.F**&F+F)F(F3!\"**&F(F)F+F)F1*&F(F1F+F)F3F+F3*$F(F1F**$F(F .\"#7F(F/*$F(F3!\"'F5F)F)F(F.F+F1,(F*F)F(F)F9F)F/,&F*F)F+F)F/,(F*F)F+F )F9F)F*,*F9F3F+F)F*F)F(F)F*" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 369 "qbw_order2:=factor((gbw-2*th_blackVertex*th_whit eVertex*C/Exp_y2_eta1_eta2-2*th_blackVertex/Exp_y2_eta1_eta2*C-2*h_whi teVertex*h_blackVertex*C/Exp_y1_eta1_eta2-2*h_blackVertex*C/Exp_y1_eta 1_eta2-2*C/W*q_nonRooted_nonRooted*g_rooted_whiteVertex*g_rooted_black Vertex-C/W*q_nonRooted_whiteVertex*g_rooted_blackVertex-C/W*q_nonRoote d_blackVertex*g_rooted_whiteVertex)*W/C);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+qbw_order2G6\",$**I%eta2GF%\"\"#I%eta1GF %F),(F*\"\"\"!\"\"F,F(F,F,,**&F*F,F(F,\"\"$F(F,F-F,F*F,F-\"\"%" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 77 "qww_order Not2:=factor((gww-B/(Exp_y2_eta1_eta2*(1-G/Exp_y2_eta1_eta2)))*W/C);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.qww_orderNot2 G6\",$*.,(I%eta1GF%\"\"\"!\"\"F*I%eta2GF%F*F*,&\"\"$F*F,F*F*F,F*F)\"\" #,(F+F*F)F**&F)F*F,F*F*F+,*F1F.F,F*F+F*F)F*F+F+" }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 57 "qbb_orderNot2:=subs(\{eta1 =eta2,eta2=eta1\},qww_orderNot2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.qbb_orderNot2G6\",$*.,(I%eta1GF%\"\"\"!\"\"F*I% eta2GF%F*F*,&\"\"$F*F)F*F*F)F*F,\"\"#,(F+F*F,F**&F)F*F,F*F*F+,*F1F.F,F *F+F*F)F*F+F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 31 "qbw_orderNot2:=factor(gbw*W/C);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.qbw_orderNot2G6\",$**I%eta1GF%\"\"\"I%et a2GF%F),**&F(F)F*F)\"\"$F*F)!\"\"F)F(F)F.,(F(F)F.F)F*F)F)\"\"%" } {TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 68 "Introduction of the algebraic series gamma1(z1,z2) and gamma2(z1,z2)" }{TEXT 204 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 274 "We make a change of var iable by introducing the variable z1 and z2 such that z1=W/y2 and z2=W /y1. These two variables will be the variables for the generating seri es of the k-rooted irreducible quadrangulations. We calculate the (eta 1,eta2)-rational expression of y1 and y2 " }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 45 "Exp_z1_eta1_eta2:=factor(W/Exp_y2 _eta1_eta2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I 1Exp_z1_eta1_eta2G6\",$*(I%eta1GF%\"\"\",(F(F)!\"\"F)I%eta2GF%F)F),&F+ F)F(F)!\"#F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 45 "Exp_z2_eta1_eta2:=factor(W/Exp_y1_eta1_eta2);" }{MPLTEXT 1 0 0 " " }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I1Exp_z2_eta1_eta2G6\",$*(I%eta2 GF%\"\"\",(I%eta1GF%F)!\"\"F)F(F)F),&F,F)F(F)!\"#F," }{TEXT 20 0 "" }} }{EXCHG {PARA 270 "" 0 "" {TEXT 206 178 "We introduce the algebraic se ries gamma1 and gamma2 such that all the generating series of k-rooted irreducible quadrangulations will be rational expressions of gamma1 a nd gamma2" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 46 "Eq_z1_gamma1_gamma2:=-z1*(1+gamma2)**2+gamma1;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I4Eq_z1_gamma1_gamma2G6\",&*&I#z1 GF%\"\"\",&F)F)I'gamma2GF%F)\"\"#!\"\"I'gamma1GF%F)" }{TEXT 20 0 "" }} }{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 46 "Eq_z2_gamma1_gamma2:=-z2* (1+gamma1)**2+gamma2;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I4Eq_z2_gamma1_gamma2G6\",&*&I#z2GF%\"\"\",&F)F)I'gamm a1GF%F)\"\"#!\"\"I'gamma2GF%F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> \+ " 0 "" {MPLTEXT 1 0 43 "Exp_z1_gamma1_gamma2:=gamma1/(1+gamma2)**2;" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 43 "Exp_z 2_gamma1_gamma2:=gamma2/(1+gamma1)**2;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I5Exp_z2_gamma1_gamma2G6\"*&I'gamma2GF%\" \"\",&F(F(I'gamma1GF%F(!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 " " {TEXT 206 83 "We give the beginning of the development of gamma1 and gamma2 in terms of z1 and z2" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> \+ " 0 "" {MPLTEXT 1 0 33 "n:=0:dev_gamma1:=0:dev_gamma2:=0:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 14 "while (n " 0 "" {MPLTEXT 1 0 90 "Eq1_gamma1_gamma2_eta1_eta 2:=factor(numer(subs(z1=Exp_z1_eta1_eta2,Eq_z1_gamma1_gamma2)));" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I " 0 "" {MPLTEXT 1 0 90 "Eq2_gamma1_gamma2_eta 1_eta2:=factor(numer(subs(z2=Exp_z2_eta1_eta2,Eq_z2_gamma1_gamma2)));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I " 0 "" {MPLTEXT 1 0 103 "Eq_gamma1_gamma2_ eta1:=factor(resultant(Eq1_gamma1_gamma2_eta1_eta2,Eq2_gamma1_gamma2_e ta1_eta2,eta2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 " 6#>I6Eq_gamma1_gamma2_eta1G6\"*&,*I'gamma1GF%!\"\"*&F(\"\"\"I%eta1GF%F +F+F,F+*&F,F+I'gamma2GF%F+F+F+,hnF*\"\"$*&F(F+F,\"\"#!\"$*$F(F0F)*&F(F 0F,F+F0*&F(F0F,F2F3*&F(F+F.F+F)*&F,F2F(F2!\"'*(F,F+F.F2F(F+F2*$F(F2!\" #*&F,F+F(F2\"\"'*(F,F0F.F+F(F+F2F(F)*&F,F0F.F+F+*&F,F0F.F2F0*&F,F0F.F0 F0*&F,F0F.\"\"%F+*&F(F+F,F0F+*&F(F2F,F0F2*(F,F2F.F0F(F+F+*(F,F0F.F0F(F +F)*&F(F0F,F0F+*(F,F0F(F2F.F+F+*(F,F0F(F2F.F2F+*(F,F2F.F+F(F2F<*(F,F2F .F2F(F2F<*(F,F+F.F+F(F2F+*(F,F+F.F2F(F2F+*(F,F2F.F+F(F+F9*(F,F2F.F2F(F +F<*(F,F+F.F+F(F+\"\"&F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 165 "We conjecture that the factor which is of degree one in eta1 is always null, and thus conjecture from it a rational expressi on of eta1 in terms of gamma1 and gamma2" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "Exp_eta1_gamma1_gamma2:=solve(-ga mma1+gamma1*eta1+eta1+eta1*gamma2,eta1);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I7Exp_eta1_gamma1_gamma2G6\"*&I'gamma1GF% \"\"\",(F'F(F(F(I'gamma2GF%F(!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 184 "It is logical by a clear argument of symmet ry to conjecture that the rational expression of eta2 in terms of gamm a1 and gamma2 is the expression of eta1 by exchanging gamma1 and gamma 2" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 83 "Exp_ eta2_gamma1_gamma2:=subs(\{gamma1=gamma2,gamma2=gamma1\},Exp_eta1_gamm a1_gamma2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I7 Exp_eta2_gamma1_gamma2G6\"*&I'gamma2GF%\"\"\",(I'gamma1GF%F(F(F(F'F(! \"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 264 "We che ck that these two rational expressions are correct by checking that th e two following expressions are null (in fact we check that the relati ons z1=W/y2 and z2=W/y1 remain true when we replace z1, z2, y1, y2 and W by their expression in terms of eta1 and eta2" }{TEXT 206 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 154 "factor(subs(beta1=eta1*(- 1+eta2)/(3*eta1*eta2-eta1-1-eta2),beta2=eta2*(-1+eta1)/(-1-eta2+3*eta1 *eta2-eta1),Exp_x1_beta1_beta2*(1+F)**2)-Exp_y1_eta1_eta2);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 154 "factor(subs(beta1 =eta1*(-1+eta2)/(3*eta1*eta2-eta1-1-eta2),beta2=eta2*(-1+eta1)/(-1-eta 2+3*eta1*eta2-eta1),Exp_x2_beta1_beta2*(1+F)**2)-Exp_y2_eta1_eta2);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 275 "" 0 "" {TEXT 204 80 "All series of k-roote d irreducible quadrangulations are (gamma1,gamma2)-rational" }{TEXT 204 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 552 "As we have seen eta 1=gamma1/(1+gamma1+gamma2) and eta2=gamma2/(1+gamma1+gamma2) when (z1, z2) and (y1,y2) are linked by the change of variable (z1,z2)=(W/y2,W/y 1). We have also seen that each series of k-rooted irreducible quadran gulations composed with (W/y2,W/y1) admits an (eta1,eta2)-rational exp ression. By substituting eta1 and eta2 respectively by gamma1/(1+gamma 1+gamma2) and gamma2/(1+gamma1+gamma2) in such an (eta1,eta2)-rational expression, we get a (gamma1,gamma2)-rational expression for the seri es of k-rooted irreducible quadrangulations" }{TEXT 206 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 111 "q_nonRooted_rooted:=facto r(subs(\{eta1=Exp_eta1_gamma1_gamma2,eta2=Exp_eta2_gamma1_gamma2\},q_n onRooted_rooted));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3q_nonRooted_rootedG6\"*.,<\"\"\"F(*&I'gamma2GF%F(I'gamma1GF% \"\"$F(*&F+F(F*F,F(*$F+F,F(*&F*\"\"#F+F0F(*&F+F(F*F(\"\"(*$F+F0F,F*F,F +F,*&F+F(F*F0\"\"&*&F+F0F*F(F5*$F*F0F,*$F*F,F(F(F+F(F*F(,*F(F(F+F0F3F( F*F(!\"\",*F+F(F(F(F*F0F7F(F:,(F+F(F(F(F*F(!\"#" }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 117 "q_nonRooted_nonRooted:=fa ctor(subs(\{eta1=Exp_eta1_gamma1_gamma2,eta2=Exp_eta2_gamma1_gamma2\}, q_nonRooted_nonRooted));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I6q_nonRooted_nonRootedG6\",$*0,dr\"\"#\"\"\"*&I'gamma 2GF%F*I'gamma1GF%\"\"'\"$*>*$F,\"\"&\"$7\"*&F,\"\"%F-F*\"$v'*&F-F*F,\" \")F1*&F,F*F-F1\"$m%*&F,F*F-F4F5*&F-F*F,F1F9*&F,\"\"$F-\"\"(\"\"**&F,F 4F-F.\"#5*&F,F*F-F>\"#[*&F,F*F-F7F1*&F,F)F-F1\"$$o*&F,F.F-F=\"#\"**&F, F)F-F7F=*&F,F)F-F>\"#P*&F,F.F-F4FA*&F,F>F-F=F?*&F,F)F-F.\"$;#*&F,F=F-F .FH*$F,F.\"#c*&F,F*F-F=\"$?'*&F,F)F-F=\"%:9*&F-F*F,F=FT*$F-F=F2*&F,F)F -F)\"$]**&F-F*F,F*\"$9\"*&F,F)F-F4\"%j7*$F-F)FR*&F,F=F-F=\"%3<*&F,F=F- F4\"%k6*&F-F=F,F4F]o*&F-F)F,F4Fhn*&F,F=F-F)FV*&F,F7F-F)F=*&F,F1F-F)FF* &F,F.F-F)FO*&F,F>F-F)FKF,\"#;F-Feo*$F-F4\"$S\"*&F,F=F-F1\"$X%*&F,F4F-F 4\"$[&*&F,F4F-F1\"$A\"*&F,F1F-F=Fio*&F,F1F-F4F]p*&F,F1F-F1FA*&F-F*F,F. F/*&F-F*F,F>FC*$F-F1F2*$F-F.FR*&F-F*F,F)\"$`$*&F-F)F,F*Ffp*$F-F>Feo*$F -F7F)*$F,F>Feo*$F,F7F)*$F,F)FR*$F,F=F2*$F,F4FgoF*F-F*F,F*,*FenF=F,!\" \"F-F`qF`qF*F`q,*F-F*F*F*F,F)F\\qF*!\"#,*F*F*F-F)FinF*F,F*Fbq,(F-F*F*F *F,F*!\"$F`q" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 115 "q_rooted_blackVertex:=factor(subs(\{eta1=Exp_eta1_gamma1_gamma2 ,eta2=Exp_eta2_gamma1_gamma2\},q_rooted_blackVertex));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I5q_rooted_blackVertexG6\"**I 'gamma2GF%\"\"#I'gamma1GF%F(,*F)\"\"\"F+F+F'F(*$F'F(F+!\"\",(F)F+F+F+F 'F+!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 115 "q_rooted_whiteVertex:=factor(subs(\{eta1=Exp_eta1_gamma1_gamma2,e ta2=Exp_eta2_gamma1_gamma2\},q_rooted_whiteVertex));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I5q_rooted_whiteVertexG6\"**I'g amma2GF%\"\"#I'gamma1GF%F(,*\"\"\"F+F)F(*$F)F(F+F'F+!\"\",(F)F+F+F+F'F +!\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 121 " q_nonRooted_blackVertex:=factor(subs(\{eta1=Exp_eta1_gamma1_gamma2,eta 2=Exp_eta2_gamma1_gamma2\},q_nonRooted_blackVertex));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I8q_nonRooted_blackVertexG6\" ,$*.I'gamma2GF%\"\"#I'gamma1GF%F),>\"\"%\"\"\"*&F(F-F*\"\"$F/*&F*F-F(F /\"\")*$F*F/\"\"&*&F(F)F*F)\"\"**&F*F-F(F-\"#M*$F*F)\"#9F(\"#;F*\"#8*& F*F-F(F)\"#H*&F*F)F(F-\"#@*$F(F)\"#C*$F(F/F:*$F(F,F,F-,*F6F/F(!\"\"F*F EFEF-FE,*F*F-F-F-F(F)F@F-!\"#,(F*F-F-F-F(F-!\"$FG" }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 121 "q_nonRooted_whiteVertex:= factor(subs(\{eta1=Exp_eta1_gamma1_gamma2,eta2=Exp_eta2_gamma1_gamma2 \},q_nonRooted_whiteVertex));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I8q_nonRooted_whiteVertexG6\",$*.I'gamma1GF%\"\"#I'gam ma2GF%F),>\"\"%\"\"\"*&F*F-F(\"\"$\"\")*&F(F-F*F/F/*$F(F/\"#;*&F*F)F(F )\"\"**&F(F-F*F-\"#M*$F(F)\"#CF*\"#8F(F3*$F(F,F,*&F(F-F*F)\"#@*&F(F)F* F-\"#H*$F*F)\"#9*$F*F/\"\"&F-,*F6F/F*!\"\"F(FEFEF-FE,*F-F-F(F)F8F-F*F- !\"#,(F(F-F-F-F*F-!\"$FG" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 95 "qbb_order2:=factor(subs(\{eta1=Exp_eta1_gamma1_gamma 2,eta2=Exp_eta2_gamma1_gamma2\},qbb_order2));" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>I+qbb_order2G6\",$*2I'gamma2GF%\"\"% I'gamma1GF%\"\"#,R\"\"$\"\"\"*$F(\"\"&F-*&F(F)F*F.\"#I*&F(F.F*F)\"#6*& F*F.F(F0F-*&F(F.F*F-\"#s*&F(F+F*F-\"#V*&F*F.F(F-\"##**$F*F-\"#Q*&F(F+F *F+\"$K\"*&F*F.F(F.\"#\")*&F(F+F*F)F-*$F*F+F9*&F(F-F*F-\"\"**&F*F+F(F) \"\"(*&F(F-F*F+\"#bF(\"#:F*\"#?*$F*F)\"#7*&F*F.F(F+\"$E\"*&F*F+F(F.\"$ F\"*$F(F+F2*$F(F-F2*$F(F)FJF.,*F@F-F(!\"\"F*FVFVF.FV,*F.F.F*F+FCF.F(F. FV,&F.F.F*F.!\"#,*F*F.F.F.F(F+FRF.FY,(F*F.F.F.F(F.!\"$FV" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 59 "qww_order2:=subs(\{g amma1=gamma2,gamma2=gamma1\},qbb_order2);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+qww_order2G6\",$*2I'gamma1GF%\"\"%I'gamm a2GF%\"\"#,R\"\"$\"\"\"*$F(\"\"&F-*&F*F.F(F)\"#I*&F*F)F(F.\"#6*&F*F.F( F0F-*&F(F.F*F-\"#s*&F*F-F(F+\"#V*&F*F.F(F-\"##**$F*F-\"#Q*&F*F+F(F+\"$ K\"*&F(F.F*F.\"#\")*&F(F+F*F)F-*$F*F+F9*&F*F-F(F-\"\"**&F*F+F(F)\"\"(* &F*F+F(F-\"#bF(\"#:F*\"#?*$F*F)\"#7*&F(F+F*F.\"$E\"*&F(F.F*F+\"$F\"*$F (F+F2*$F(F-F2*$F(F)FJF.,*F@F-F*!\"\"F(FVFVF.FV,*F(F.F.F.F*F+FCF.FV,&F. F.F*F.!\"#,*F.F.F(F+FRF.F*F.FY,(F(F.F.F.F*F.!\"$FV" }{TEXT 20 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 95 "qbw_order2:=factor(subs(\{ eta1=Exp_eta1_gamma1_gamma2,eta2=Exp_eta2_gamma1_gamma2\},qbw_order2)) ;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+qbw_order2 G6\",$**I'gamma2GF%\"\"#I'gamma1GF%F),**&F*\"\"\"F(F-\"\"$F(!\"\"F*F/F /F-F/,(F*F-F-F-F(F-!\"$!\"%" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 101 "qbb_orderNot2:=factor(subs(\{eta1=Exp_eta1_gamma 1_gamma2,eta2=Exp_eta2_gamma1_gamma2\},qbb_orderNot2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.qbb_orderNot2G6\",$*.I'ga mma2GF%\"\"#I'gamma1GF%\"\"\",(F*\"\"%\"\"$F+F(F.F+,**&F*F+F(F+F.F(!\" \"F*F1F1F+F1,*F+F+F*F)*$F*F)F+F(F+F1,(F*F+F+F+F(F+F1F1" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 101 "qww_orderNot2:=factor (subs(\{eta1=Exp_eta1_gamma1_gamma2,eta2=Exp_eta2_gamma1_gamma2\},qww_ orderNot2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I .qww_orderNot2G6\",$*.I'gamma1GF%\"\"#I'gamma2GF%\"\"\",(F(\"\"$F-F+F* \"\"%F+,**&F(F+F*F+F-F*!\"\"F(F1F1F+F1,*F(F+F+F+F*F)*$F*F)F+F1,(F(F+F+ F+F*F+F1F1" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 101 "qbw_orderNot2:=factor(subs(\{eta1=Exp_eta1_gamma1_gamma2,eta2=Exp _eta2_gamma1_gamma2\},qbw_orderNot2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I.qbw_orderNot2G6\",$**I'gamma1GF%\"\"\"I' gamma2GF%F),(F(F)F)F)F*F)!\"\",**&F(F)F*F)\"\"$F*F,F(F,F,F)F,!\"%" } {TEXT 20 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 140 "The following (gamma1,gamma2)-rational expression of the series of rooted irreducib le quadrangulations is taken from Mullin and Schellenberg" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 146 "q:=factor(Exp_z1_ gamma1_gamma2*Exp_z2_gamma1_gamma2*(1/(1+Exp_z1_gamma1_gamma2)+1/(1+Ex p_z2_gamma1_gamma2)-1)-gamma1*gamma2/(1+gamma1+gamma2)**3);" } {MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"qG6\",$*2I'ga mma1GF%\"\"$I'gamma2GF%F),4F(!\"#!\"\"\"\"\"*$F(\"\"#F-*&F(F.F*F.F-*&F (F0F*F.F.*&F(F.F*F0F.*$F*F0F-F*F,*&F*F0F(F0F.F.,&F.F.F*F.F,,&F.F.F(F.F ,,*F(F.F.F.F*F0F4F.F-,*F.F.F(F0F/F.F*F.F-,(F(F.F.F.F*F.!\"$F-" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 285 "" 0 "" {TEXT 215 97 "Obtaining the init ial coefficients counting unrooted 3-connected maps with i vertices an d j faces" }{TEXT 215 0 "" }}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 93 "Ob taining the development in z1 and z2 of the series of k-rooted irreduc ible quadrangulations" }{TEXT -1 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 259 "From the rational expressions of the k-rooted simple qu adrangulations in terms of gamma1 and gamma2 and from the developments of gamma1 and gamma2 in terms of z1 and z2, we obtain developments of the k-rooted irreducible quadrangulations in terms of z1 and z2" } {TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "dev_q:=e xpand(series(subs(gamma1=dev_gamma1,gamma2=dev_gamma2,q),t,N+1)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 107 "dev_ q_nonRooted_rooted:=expand(series(subs(gamma1=dev_gamma1,gamma2=dev_ga mma2,q_nonRooted_rooted),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 113 "dev_q_nonRooted_nonRooted:=expand(seri es(subs(gamma1=dev_gamma1,gamma2=dev_gamma2,q_nonRooted_nonRooted),t,N +1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 111 "dev_q_rooted_blackVertex:=expand(series(subs(gamma1=dev_gamma1,ga mma2=dev_gamma2,q_rooted_blackVertex),t,N+1)):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 111 "dev_q_rooted_whiteVertex: =expand(series(subs(gamma1=dev_gamma1,gamma2=dev_gamma2,q_rooted_white Vertex),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 117 "dev_q_nonRooted_blackVertex:=expand(series(subs(gamm a1=dev_gamma1,gamma2=dev_gamma2,q_nonRooted_blackVertex),t,N+1)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 117 "dev_ q_nonRooted_whiteVertex:=expand(series(subs(gamma1=dev_gamma1,gamma2=d ev_gamma2,q_nonRooted_whiteVertex),t,N+1)):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "dev_qbb_order2:=expand(ser ies(subs(gamma1=dev_gamma1,gamma2=dev_gamma2,qbb_order2),t,N+1)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "dev_q ww_order2:=expand(series(subs(gamma1=dev_gamma1,gamma2=dev_gamma2,qww_ order2),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "dev_qbw_order2:=expand(series(subs(gamma1=dev_gamma1, gamma2=dev_gamma2,qbw_order2),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 97 "dev_qbb_orderNot2:=expand(series( subs(gamma1=dev_gamma1,gamma2=dev_gamma2,qbb_orderNot2),t,N+1)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 97 "dev_q ww_orderNot2:=expand(series(subs(gamma1=dev_gamma1,gamma2=dev_gamma2,q ww_orderNot2),t,N+1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 97 "dev_qbw_orderNot2:=expand(series(subs(gamma1=dev_g amma1,gamma2=dev_gamma2,qbw_orderNot2),t,N+1)):" }{MPLTEXT 1 0 0 "" }} }}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 59 "Apply Burnside-Liskovets form ula to obtain the enumeration " }{TEXT -1 0 "" }}{EXCHG {PARA 270 "" 0 "" {TEXT 206 90 "Burnside-Liskovets formula can be easily adapted to \+ give the following equation on series " }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 683 "sum_k_rooted_3connected_two_vari ables:=convert(expand(series(dev_q+z1/z2*subs(z1=z1**2,z2=z2**2,t=t**2 ,dev_qbb_order2)+z2/z1*subs(z1=z1**2,z2=z2**2,t=t**2,dev_qww_order2)+s ubs(z1=z1**2,z2=z2**2,t=t**2,dev_qbw_order2)+t*z1*subs(z1=z1**2,z2=z2* *2,t=t**2,dev_q_rooted_whiteVertex+dev_q_nonRooted_whiteVertex)+t*z2*s ubs(z1=z1**2,z2=z2**2,t=t**2,dev_q_rooted_blackVertex+dev_q_nonRooted_ blackVertex)+t**2*z1*z2*subs(z1=z1**2,z2=z2**2,t=t**2,dev_q_nonRooted_ rooted+dev_q_nonRooted_nonRooted)+sum(phi(k)*(z1/z2*subs(z1=z1**k,z2=z 2**k,t=t**k,dev_qbb_orderNot2)+z2/z1*subs(z1=z1**k,z2=z2**k,t=t**k,dev _qww_orderNot2)+subs(z1=z1**k,z2=z2**k,t=t**k,dev_qbw_orderNot2)),k=3. .N),t,N+1)),polynom):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 192 "By dividing each coefficient by 2*(i+j), we get enumera tion of unrooted 3-connected maps. The coefficient of z1^i*z2^j is the number of unrooted 3-connected map with i+1 vertices and j+1 faces" } {TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 122 "non_roo ted_3connected_map_two_variables:=expand(factor(series(1/2*int(sum_k_r ooted_3connected_two_variables/t,t),t,N+1)));" }{MPLTEXT 1 0 0 "" }} {PARA 273 "" 1 "" {XPPMATH 20 "6#>IHnon_rooted_3connected_map_two_vari ablesG6\"+?I\"tGF%*&I#z2GF%\"\"$I#z1GF%F*\"\"'*&F)\"\"%F+F.\"\"),&*&F) F.F+\"\"&\"\"\"*&F)F2F+F.F3\"\"*,$*&F)F2F+F2F*\"#5,&*&F)F2F+F,\"\"#*&F )F,F+F2F;\"#6,(*&F)F2F+\"\"(F;*&F)F,F+F,F=*&F)F@F+F2F;\"#7,&*&F)F,F+F@ \"#;*&F)F@F+F,FF\"#8,(*&F)F,F+F/F8*&F)F/F+F,F8*&F)F@F+F@\"#p\"#9,**&F) F/F+F@\"$F\"*&F)F5F+F,F,*&F)F@F+F/FQ*&F)F,F+F5F,\"#:,(*&F)F5F+F@\"$G\" *&F)F@F+F5FX*&F)F/F+F/\"$T&FF,**&F)F5F+F/\"%)=\"*&F)F8F+F@\"#g*&F)F/F+ F5Fhn*&F)F@F+F8Fjn\"#<,,*&F)F=F+F@F]o*&F)F8F+F/\"%T9*&F)F@F+F=F]o*&F)F 5F+F5\"%'4&*&F)F/F+F8Fao\"#=,**&F)F/F+F=\"%K5*&F)F=F+F/Fio*&F)F5F+F8\" &#)>\"*&F)F8F+F5F\\p\"#>,,*&F+FCF)F/\"$'Q*&F+F=F)F5\"&ls\"*&F+F5F)F=Fc p*&F+F/F)FCFap*&F+F8F)F8\"&'e]\"#?" }{TEXT 20 0 "" }}}}}}}{SECT 1 {PARA 286 "" 0 "" {TEXT 214 62 "Fast enumeration using algebraicity of series of k-rooted maps" }{TEXT 214 0 "" }}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 44 "Enumeration according to the number of edges" }{TEXT 207 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 153 "To perform the trea tment of this part, you have either to have a recent version of maple \+ (at least version 9) and use the command with(gfun) as follows. " } {TEXT 223 0 "" }{TEXT 223 300 "\nYou can also download algolib for you r version of maple at the following adress http://algo.inria.fr/librar ies/. Then, before the instruction with(gfun); you add the instruction libname:=\"PATH\"; where PATH is the path of the repertory where you \+ have installed algolib (see README at the web adress)." }{TEXT 223 0 " " }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }{MPLTEXT 1 0 0 "" } }{PARA 272 "" 1 "" {TEXT 230 61 "Warning, the assigned name rectoproc \+ now has a global binding" }{TEXT 230 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#7PI(LaplaceG6\"I.algebraicsubsGF%I.algeqtodiffeqGF%I.al geqtoseriesGF%I.algfuntoalgeqGF%I&borelGF%I.cauchyproductGF%I.diffeq*d iffeqGF%I.diffeq+diffeqGF%I,diffeqtableGF%I2diffeqtohomdiffeqGF%I,diff eqtorecGF%I)guesseqnGF%I(guessgfGF%I0hadamardproductGF%I0holexprtodiff eqGF%I)invborelGF%I,listtoalgeqGF%I-listtodiffeqGF%I0listtohypergeomGF %I+listtolistGF%I.listtoratpolyGF%I*listtorecGF%I-listtoseriesGF%I,max degcoeffGF%I*maxdegeqnGF%I,maxordereqnGF%I,mindegcoeffGF%I*mindegeqnGF %I,minordereqnGF%I*optionsgfGF%I,poltodiffeqGF%I)poltorecGF%I/ratpolyt ocoeffGF%I(rec*recGF%I(rec+recGF%I,rectodiffeqGF%I,rectohomrecGF%I*rec toprocGF%I.seriestoalgeqGF%I/seriestodiffeqGF%I2seriestohypergeomGF%I- seriestolistGF%I0seriestoratpolyGF%I,seriestorecGF%I/seriestoseriesGF% " }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 16 "with(n umtheory);" }}{PARA 272 "" 1 "" {TEXT 230 68 "Warning, the protected n ame order has been redefined and unprotected" }{TEXT 230 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#7QI&GIgcdG6\"I)bigomegaGF%I&cfracGF%I)cfrac polGF%I+cyclotomicGF%I)divisorsGF%I)factorEQGF%I*factorsetGF%I'fermatG F%I)imagunitGF%I&indexG6$I*protectedGF1I(_syslibGF%I/integral_basisGF% I)invcfracGF%I'invphiGF%I*issqrfreeGF%I'jacobiGF%I*kroneckerGF%I'lambd aGF%I)legendreGF%I)mcombineGF%I)mersenneGF%I(migcdexGF%I*minkowskiGF%I (mipolysGF%I%mlogGF%I'mobiusGF%I&mrootGF%I&msqrtGF%I)nearestpGF%I*nthc onverGF%I)nthdenomGF%I)nthnumerGF%I'nthpowGF%I&orderGF1I)pdexpandGF%I$ phiGF%I#piGF%I*pprimrootGF%I)primrootGF%I(quadresGF%I+rootsunityGF%I*s afeprimeGF%I&sigmaGF%I*sq2factorGF%I(sum2sqrGF%I$tauGF%I%thueGF%" } {TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 248 "Here we can i ncrease the number of edges until which we want to perform the enumera tion of unrooted maps. Indeed we use methods from computer algebra to extract the initial coefficients of series of k-rooted quadrangulatio ns is far more efficient." }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 7 "N:=150;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"NG6\"\"$]\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 105 "We concentrate on the enumeration of unrooted 2-co nnected maps and, above all, unrooted 3-connected maps." }{TEXT 223 0 "" }}}{SECT 0 {PARA 281 "" 0 "" {TEXT 218 40 "Enumeration of unrooted \+ 2-connected maps" }{TEXT 218 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }{MPLTEXT 1 0 0 "" }}{PARA 272 "" 1 "" {TEXT 230 61 "Warning, the assigned name rectoproc now has a global bi nding" }{TEXT 230 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#7PI(LaplaceG 6\"I.algebraicsubsGF%I.algeqtodiffeqGF%I.algeqtoseriesGF%I.algfuntoalg eqGF%I&borelGF%I.cauchyproductGF%I.diffeq*diffeqGF%I.diffeq+diffeqGF%I ,diffeqtableGF%I2diffeqtohomdiffeqGF%I,diffeqtorecGF%I)guesseqnGF%I(gu essgfGF%I0hadamardproductGF%I0holexprtodiffeqGF%I)invborelGF%I,listtoa lgeqGF%I-listtodiffeqGF%I0listtohypergeomGF%I+listtolistGF%I.listtorat polyGF%I*listtorecGF%I-listtoseriesGF%I,maxdegcoeffGF%I*maxdegeqnGF%I, maxordereqnGF%I,mindegcoeffGF%I*mindegeqnGF%I,minordereqnGF%I*optionsg fGF%I,poltodiffeqGF%I)poltorecGF%I/ratpolytocoeffGF%I(rec*recGF%I(rec+ recGF%I,rectodiffeqGF%I,rectohomrecGF%I*rectoprocGF%I.seriestoalgeqGF% I/seriestodiffeqGF%I2seriestohypergeomGF%I-seriestolistGF%I0seriestora tpolyGF%I,seriestorecGF%I/seriestoseriesGF%" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 407 "The method consists in extracing quic kly the initial coefficients of the series of k-rooted simple quadrang ulations. First , we obtain an algebraic equation for these series. Th en, from the algebraic equation, a linear differential equation verifi ed by the series is computed. Then, taking coefficient [y^n] in this e quation, we obtain a linear recursive equation verified by the coeffic ients of the series." }{TEXT 223 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 149 "We take here the example of the series gvv. First, as g vv is eta-rational, we extract from this relation an (algebraic) equat ion linking gvv and eta " }{TEXT 223 0 "" }}}{EXCHG {PARA 287 "> " 0 " " {MPLTEXT 1 -1 29 "Eq_eta_y := -y+eta*(1-eta)^2;" }{MPLTEXT 1 -1 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I)Eq_eta_yG6\",&I\"yGF%!\"\"*&I$e taGF%\"\"\",&F+F+F*F(\"\"#F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 287 "> " 0 "" {MPLTEXT 1 -1 32 "Eq_gvv_eta:=2*eta+gvv*(3*eta-1);" }{MPLTEXT 1 -1 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+Eq_gvv_etaG6\",&I$etaGF% \"\"#*&I$gvvGF%\"\"\",&F'\"\"$!\"\"F+F+F+" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 134 "Taking the resultant of this equation with Eq_eta_y according to eta, we get an algebraic equation for gvv \+ according to the variable y" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 53 "Eq_gvv_y:=factor(resultant(Eq_gvv_eta,Eq_eta_y,e ta));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I)Eq_gvv _yG6\",0*$I$gvvGF%\"\"$\"\"%*$F(\"\"#\"\")I\"yGF%!\")*&F.\"\"\"F(F1!#O *&F.F1F(F,!#aF(F**&F.F1F(F)!#F" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 104 "From this algebraic equation, we get a linear diff erential equation by using the function algeqtodiffeq " }{TEXT 223 0 " " }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 51 "deq_gvv:=algeqtodiffe q(Eq_gvv_y,gvv(y),\{gvv(0)=0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 154 "Then taking coefficient [y^n] in the differ ential equation, we get a recurrence verified b the coefficients of gv v. This is done by the method diffeqtorec" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 42 "req_gvv:=diffeqtorec(deq_gvv,gvv( y),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 309 "From the recurrence we get a procedure enumerating the N first co efficients of gvv. This is done with the method rectoproc. We call thi s procedure with parameter N as argument to get the list of the N firs t coefficients of gvv. Then we get from this list the series dev_gvv o f the development of gvv till y^N " }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 39 "proc_gvv:=rectoproc(req_gvv,u(m),list): " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 12 "pr oc_gvv(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "dev_gvv:=sum('%[i]*y**(i-1)','i'=1..N+1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 45 "We determine the development of gfv similarly" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 28 "Eq_gfv_eta:=2+gfv*(3*eta-1);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I+Eq_gfv_etaG6\",&\"\"#\"\"\"*&I$gfvGF%F(,&I$etaGF%\"\"$!\"\"F( F(F(" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 53 "Eq _gfv_y:=factor(resultant(Eq_gfv_eta,Eq_eta_y,eta)):" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 51 "deq_gfv:=algeqtodiffe q(Eq_gfv_y,gfv(y),\{gfv(0)=2\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 42 "req_gfv:=diffeqtorec(deq_gfv,gfv(y),u(m )):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 39 "proc_gfv:=rectoproc(req_gfv,u(m),list):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 12 "proc_gfv(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "dev_gfv:=sum('%[ i]*y**(i-1)','i'=1..N+1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 35 "We determine the development of gff" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 39 "Eq_gff_eta:=-1+gff*((3 *eta-1)*(eta-1)):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 53 "Eq_gff_y:=fa ctor(resultant(Eq_gff_eta,Eq_eta_y,eta)):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 51 "deq_gff:=algeqtodiffeq(Eq_ gff_y,gff(y),\{gff(0)=1\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 42 "req_gff:=diffeqtorec(deq_gff,gff(y),u(m)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 39 "proc_ gff:=rectoproc(req_gff,u(m),list):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 12 "proc_gff(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 41 "dev_gff:=sum('%[i]*y**(i-1)','i'=1..N+1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 33 "We determ ine the development of g" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 " " {MPLTEXT 1 0 27 "Eq_g_eta:=-g+eta*(2-3*eta):" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 49 "Eq_g_y:=factor(resultant(Eq_g_eta,Eq_eta_y,eta)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 43 "deq_g :=algeqtodiffeq(Eq_g_y,g(y),\{g(0)=0\}):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 36 "req_g:=diffeqtorec(deq_g,g (y),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 35 "proc_g:=rectoproc(req_g,u(m),list):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 10 "proc_g(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 39 "dev_g:=sum('%[i]*y**(i-1)','i'=1..N+1): " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 157 "We det emine the number of unrooted 2-connected maps. First we determine the \+ series of 2nc_n where c_n is the number of unrooted 2-connected maps w ith n edges" }{TEXT 223 0 "" }}}{EXCHG {PARA 288 "> " 0 "" {MPLTEXT 1 213 134 "sum_k_rooted_2connected:=series(dev_g+y*subs(y=y**2,dev_gfv)+ y**2*subs(y=y**2,dev_gff)+sum(phi(k)*subs(y=y**k,dev_gvv),k=2..N),y,N+ 1):" }{MPLTEXT 1 213 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 182 "W e divide each coefficient of the preceding series by 2*n to achieve th e Burnside Lemma and obtain the first values of non-rooted 2-connected maps with respect to the number of edges" }{TEXT 206 0 "" }}}{EXCHG {PARA 288 "> " 0 "" {MPLTEXT 1 213 92 "non_rooted_2connectedmaps:=seri es(1/2*int(series(sum_k_rooted_2connected/y,y,N+1),y),y,N+1);" } {MPLTEXT 1 213 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I:non_rooted_2 connectedmapsG6\"+[^lI\"yGF%\"\"#\"\"\"F)F(F(\"\"$F*\"\"%\"\"'\"\"&\"# ;F,\"#U\"\"(\"$^\"\"\")\"$'f\"\"*\"%0E\"#5\"&)47\"#6\"&m\"f\"#7\"'%o(H \"#8\"(!fQ:\"#9\"(y!4\")\"#:\")^nZVF.\"*U\\ZO#\"#<\"+T4o-8\"#=\"+iB%oD (\"#>\",$GzH$3%\"#?\"-5$=%Q=B\"#@\".S\"#I\"5y(z%)yVez#[6\"#J\"5z7[@**G\"*HH p\"#K\"6Q0N:&R460'>%\"#L\"7:'[v#f[[CC\\D\"#M\"8=2_gpdAPANb\"\"#N\"8&ze #4B=4#3w%\\*\"#O\"9QHfY'oe$zI!*=e\"#P\":khwRS%zxC;SvN\"#Q\";_vi'G\"fT? m')H-A\"#R\"<^4[mX#)GWT#Hpf8\"#S\"9T\"H#*e'o'3)eGo8C$\"#V\"?5+v(Ga*4s>8Ncz 7ZeSe7\"#X\"@#)>TTe))pf@-7EM='y\"#Y\"A)4ARH%4PJoCn&zr)>\\\"#Z\"B)z'oz7 l9;o!GoBNv$3$\"#[\"C)\\!HTB]A>p(Q2&*=ae$>\"#\\\"D+>bpH]EaT6@>y>Uq@\"\" #]\"D)\\(Q<36iw\"#^\"EO<-y(y`ph%HYDK?.\\I[\"#_\"F#47!f&oGwRN \">_(4e#z#\\I\"#`\"Ggj=2#oV\"*=@IF>\"#a\"HM2Rrt:(3\"4>r@Zg&))=j >7\"#b\"H%p[wXmE!3TI*Gr'Qa?vqs(\"#c\"I;(>=nrFVt9:aG+UR9p5!\\\"#d\"J;Bi )yYFC0716/c\\FTR*>6$\"#e\"KA)fs/02.fm?>\"[*GjJdu!y>\"#f\"LqVMkP&G`(>$>>DaD(e6JP^bHV,=*\\jg)HB LLf\"#s\"V9v%e6X=gQK(\\&Q#RD)\\_!Ryd(4Le:Q\"#t\"W>JF`$eYg'e\\m!\\JQ9$Q r])QtJjL`X#\"#u\"X[dhy+rEI[XvSX!)4)ejybAo5E$G5e\"\"#v\"YuXU=wB_4h(R47^ @Jp'3$e\"p9*GP(o=5\"#w\"Y/#\\^)Qa%e.,#QK5#3DiL1[CyF\"*[2wc'\"#x\"ZW@UW $3'f>(>d#4T04D&p`HHEW()3ytOU\"#y\"enq$o5#o1b=B$3W_E_,$y\"3MmLp5*QjoMF \"#z\"fn\"RX'\\p@?;Q7C^=#)\\(QiaF\"o7Gahw\"\"#!)\"gnY/Wx3B?*yB&[+:eby^CYe#H&42?GJx%\"#$)\"in*4/U*=:;\"Qi:`Z]p(*H>x d@Y[d9#[3,?*3$\"#%)\"jn)f69nPzGvz+eV%4?())R?26M-Bc62T*[.+#\"#&)\"[oKB_ 1mmb-A!=%G%Gb8-c*QC2b>A./<]p\"fH\"\"#')\"[oSwd'3*o()eX#p\\\"pi*HdCqWun pQ!>DDfs`*R)\"#()\"\\oIn&f#GsQ[rLszA0$3%*3<`@zpd?B]o#zEnW&\"#))\"]oK)> #ogT'\\(fVrv'HOw%4h#*=)oTM\"e^$3XhmN`$\"#*)\"^om$R$o%>%4txP6z^ab%fAVEy '=W=>=._e=AMH#\"#!*\"_ogd,!yI_!*)QM%=(GgAB7q^;9x@!))yU7vVIn\"*[\"\"#\" *\"_oU44Sh>N!3vD9uEL\")GuIe+lgUZ@b#z'z;)et'*\"##*\"`o;*))))>Nln?/2#e*= W=35)*pPP`k)>>F9z$*4\\lG'\"#$*\"aoa[yD7gbMA3T4)[\"pHM>>jJ=)HbYTlXz$[f3 (3%\"#%*\"boAC*zgg:\")Qs'*)zh;w0ask1Om)=w&3FhqZaY.#eE\"#&*\"coq1b$py7^ t_zFsK!G6fM)4c3SVgn$HvT2NQcaH<\"#'*\"doo\"[!f?@P:\"HY9SwLfw;m.Icj1n,dh Kxf\"=E\\d7\"\"#(*\"do9&eOQ8PW&ypizGETh.lk!o8HxK!ek+]!\\Hc[;It\"#)*\"e oGB&=[#zzV*y$QZzNl#H727`akR2jHa9r'H'yG(ouZ\"#**\"fo3N\">z*)*)GDydl)y&* GhY.%*yC6D9*yk?zCJx*[UH76$\"$+\"\"gog[3K0Qf\"yuV*z,mL[FoV3@v$[qT5sG(op zpZ'=!G?\"$,\"\"hoM@0Bv[bQj.i0G?V;+E>glVXGzQ/Wgu#=[/EjRA8\"$-\"\"ho9(o %\\-A)y'\\3_u80Oe+k4_+RRe@^U\"yJH,;dFgxD')\"$.\"\"io!)yXO;W(pOStdE=D5L Uq2aub+$pDEri1Yp,J?$>$Gc\"$/\"\"jo!o$p$)>$fG_^`rd]&QJ:>:U2148JA(eT&R!R wZ**\\KntO\"$0\"\"[p_hdA\"yt#*)yoz#3O:)z&GVyJ%)\\E?I[aC\"3!*z$>v+(oh)R #\"$1\"\"\\pyg&>(GW)*3!zt$4a'>*[RTel([oApzSbph&HR\"4!*[_mfm:\"$2\"\"]p +c/&zMF-#yDT15#f03>!oj1liY[(y[p.\\y'e\"HF(=y\\B5\"$3\"\"]psX[wYv7L\"*= !zHs4$>rJJ=K`;X()\\,o2>sMPh(QFg!z)o'\"$4\"\"^pIFF-h\"H]]U7*ecqf]]A-Hn_ j^h,.2pi#eb(G@d!G1dsV\"$5\"\"_p]+aVh&p_ee4s\"GtvfJ&e1F+&e$4^3b0,lv%[o] t>S5DfG\"$6\"\"`pmlxk6@NAahJ'pJ6,%))=0BR'y1Rmm?EtWO!\\N\"*z1Qo:-(=\"$7 \"\"apC-G7Z28#f)*))4A#QE!Q/IbF5h2OD+JDSMy'GOkv&=#*oPOA\"\"$8\"\"aps2>* eJJIWNOV\"[kz2u\"y^#fx\"3d@!G,zIg)*fBF&)p?Ec<3!)\"$9\"\"bpu/r[Imf+;YJ' zm1JMxo@!4S?mXM%*R$G(R`7kJ:.&y[-UU_\"$:\"\"cpO9=#=&)4D%*Rqi,kz!3>zc\"[ n8fY=UR`jhqyX\\L?#GKglxKM\"$;\"\"dp'**Gx3AQ97.?]orD4u07Cb0`\\$[?kJbUO` lSwF(o=\"\"fp!\\\"=>PeVPm/<[_@OQ27MdRi$fXb!\\-pSZ%>=)Gy?SK'p&3#>4j??MHb#*3$RnxuJ7%fV7;i[yI=.bw$[T\"$@\"\"hp!=+rIP \\[KU!>BMS['4C#3>8$z))oVu(4\"HYes>7EKe***fJ xnj]UMGair%*[DO\\w,%=m&z6J$\"$F\"\"]qVT$3_))z&o_2x92>EDlA]$>'R#QYkJ*zc 0RGulQ\\m(z$Rr/rP5[Su@\"$G\"\"^q1P&)oF-h9vcntu@4c@\"4jPh$)R;$R0^Xxkp9M Mo_Zm'Q*41&37s?G9\"$H\"\"^qv'=$QN;<%[$)[S)\\W^0^k.S%fpp)Q/&)oN;)3tS#Gl ZNB\\@R7&zJ!GGQ*\"$I\"\"_qeKZ([pk.@\"f0H2O\">9tk&eP!e=8FJ.,\")4(e)fV&p Ozx*fl:'4e$yva;'\"$J\"\"`qDy$QC'f%ek90XE>D;*\\6#)*=$=]8b;h6SC;-+D0ni&3 gw@0%\"$K\"\"aqM#>%H:hwGeM0(=SN79slq8M&\\)3#=F)=`ap#e) [v$3z@))*3wPm#\"$L\"\"bq!eE9Kn\"[F,gZk\"\\#4Wnq'3qz\\]pEbCH[[*Qb$3Ft,B bi!eVV,N!HAV^<\"$M\"\"cqe*QyEGNMnAk\\WOh0h* G9*y1x\"p\"z^6\"$N\"\"cqTiZ(H2JKay#ReLF9!pL!4qO5.N(zSe#>Tuz:l1U%=pgI)4 /2kJVh/(fd(\"$O\"\"dq=g(*p]53,_6K!4myg+U%H\"HGi>LwToohw()>h6_(H8p$*R#p S[m'fE6\"3%)\\\"$P\"\"eqjXg19R!4qjN`D@[*Q[L%pfJ*4'*\\ZYGxAIoRqRb!pnmA6 <*)4Ew$*p@U&zK\"$Q\"\"fq=y4p'o]9b`Icw2pD4n*4=*oXR#eYg)4rppsg#G,zvtm3) \\1dxO2;,![$e@\"$R\"\"gqIPqL=^=Q7$[1\"o#>y;fMCil1&Qfdi\"\\)orm*4qF>S[w 9'eWU40&R=AbKK4>W)[$[!=6w*>Q7 J&pBx>zg>F(>l5\\`$*\"$T\"\"hqys^HIQ8Q\\)*)[Wu'3SAjL/7o()4uC4q<'o(=^&f@ Bx.`$)\\>\")[&3()=(pHb2fh\"$U\"\"iqgl*4fn2O#ep/mT)>`nMV5)>#)H'oSYp[7* \\WEWl^e\\Lt=7:C-VDuM=]sKcS\"$V\"\"jqz'*QfJ7@e&=79b`>e(>d8%p*GM&GfaS6$ e='z,1*)[&zu[\\&)p>$*G/x$yX1)H>n#\"$W\"\"[r#=_%oaa!Q\\tgU0?TD^e$eV>6?A*f6\"$Y\"\"\\r5O2OZ/g5T5W:Vn$ zk;A#)[E[NO*3(QAI)z]=@hsv`!Ha*ogp9=K=(4i?fcJWw\"$Z\"\"]rp.EA*QC@\"Ro') 3tl;q,2H874Ol60QL;x#yF_N!QUZ+wc**>*)>vZ48*e4=F0(Q]\"$[\"\"^rUg_#HlY&H \"3&R!)*y/oFIA<,SM#HtCX*erC=AO3\\5s!R1\")=7kUJ(4yxx%y!o_w@L\"$\\\"\"_r q!*4'*>Eg8\\:S*\\#4)RaSrz<%>3%*)HzqWK]gN3w_,u:%p4-0z6L\"f&>#>mmhW>-># \"$]\"-I\"OGI*protectedGFi]l6#F)F1" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 281 "" 0 "" {TEXT 218 40 "Enumeration of unrooted 3-connected maps" } {TEXT 218 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 210 "The same trea tment is performed as for the case of the fast enumeration of unrooted 2-connected maps. Here the variable z and the series gamma play respe ctively the role that y and eta play for 2-connected maps" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 34 "Eq_gamma_z:=-z*(1+ gamma)**2+gamma;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 " 6#>I+Eq_gamma_zG6\",&*&I\"zGF%\"\"\",&F)F)I&gammaGI*protectedGF,F)\"\" #!\"\"F+F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 65 "W e determine the development of q_rootedFace_nonRootedFace_order2" } {TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 120 "Eq_q_ro otedFace_nonRootedFace_order2_gamma:=-q_rf_nrf*((1+3*gamma+gamma^2)*(2 *gamma+1)^2)+(1+3*gamma+3*gamma^2)*gamma^2:" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 119 "Eq_q_rootedFace_nonRootedFace_order2_z:=factor(resu ltant(Eq_q_rootedFace_nonRootedFace_order2_gamma,Eq_gamma_z,gamma)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 121 "deq _q_rootedFace_nonRootedFace_order2:=algeqtodiffeq(Eq_q_rootedFace_nonR ootedFace_order2_z,q_rf_nrf(z),\{q_rf_nrf(0)=0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 107 "req_q_rootedFace_nonR ootedFace_order2:=diffeqtorec(deq_q_rootedFace_nonRootedFace_order2,q_ rf_nrf(z),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 99 "proc_q_rootedFace_nonRootedFace_order2:=rectoproc(req _q_rootedFace_nonRootedFace_order2,u(m),list):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 42 "proc_q_rootedFace_nonRootedFace_order2(N):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 71 "dev_q _rootedFace_nonRootedFace_order2:=sum('%[i]*z**(i-1)','i'=1..N+1):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 61 "We determi ne the development of q_nonRootedFace_nonRootedFace" }{TEXT 223 0 "" } }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 160 "Eq_q_nonRootedFace_nonR ootedFace_order2_gamma:=2*gamma^2*(1+5*gamma+10*gamma^2+9*gamma^3)*(1+ gamma)^2+q_nrf_nrf*((gamma-1)*(1+3*gamma+gamma^2)^2*(2*gamma+1)^3):" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 125 "Eq_q_nonRootedFace_nonRootedFace _order2_z:=factor(resultant(Eq_q_nonRootedFace_nonRootedFace_order2_ga mma,Eq_gamma_z,gamma)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 129 "deq_q_nonRootedFace_nonRootedFace_order2:=algeqt odiffeq(Eq_q_nonRootedFace_nonRootedFace_order2_z,q_nrf_nrf(z),\{q_nrf _nrf(0)=0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 114 "req_q_nonRootedFace_nonRootedFace_order2:=diffeqtore c(deq_q_nonRootedFace_nonRootedFace_order2,q_nrf_nrf(z),u(m)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 97 "req_q _nonRootedFace_nonRootedFace_order2:=subs(_C[0]=2,req_q_nonRootedFace_ nonRootedFace_order2):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 105 "proc_q_nonRootedFace_nonRootedFace_order2:=rectop roc(req_q_nonRootedFace_nonRootedFace_order2,u(m),list):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 45 "proc_q_nonRooted Face_nonRootedFace_order2(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 74 "dev_q_nonRootedFace_nonRootedFace_order2:=s um('%[i]*z**(i-1)','i'=1..N+1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 58 "We determine the development of q_rootedFace _vertex_order2" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 90 "Eq_q_rootedFace_vertex_order2_gamma:=2*gamma^4-q_rf_v *((1+3*gamma+gamma^2)*(2*gamma+1)^2):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 105 "Eq_q_rootedFace_vertex_order2_z:=factor(resultant(Eq_q_rooted Face_vertex_order2_gamma,Eq_gamma_z,gamma)):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 103 "deq_q_rootedFace_vertex_o rder2:=algeqtodiffeq(Eq_q_rootedFace_vertex_order2_z,q_rf_v(z),\{q_rf_ v(0)=0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "req_q_rootedFace_vertex_order2:=diffeqtorec(deq_q_rootedFace_v ertex_order2,q_rf_v(z),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 85 "proc_q_rootedFace_vertex_order2:=rectoproc( req_q_rootedFace_vertex_order2,u(m),list):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 35 "proc_q_rootedFace_vertex_o rder2(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 64 "dev_q_rootedFace_vertex_order2:=sum('%[i]*z**(i-1)','i'=1..N+1 ):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 61 "We de termine the development of q_nonRootedFace_vertex_order2" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 139 "Eq_q_nonRootedFac e_vertex_order2_gamma:=4*(1+gamma)*(8*gamma^2+13*gamma+4)*gamma^4+q_nr f_v*((gamma-1)*(1+3*gamma+gamma^2)^2*(2*gamma+1)^3):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 111 "Eq_q_nonRootedFace_vertex_order2_z:=factor(resu ltant(Eq_q_nonRootedFace_vertex_order2_gamma,Eq_gamma_z,gamma)):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 127 "deq_ q_nonRootedFace_vertex_order2:=algeqtodiffeq(Eq_q_nonRootedFace_vertex _order2_z,q_nrf_v(z),\{q_nrf_v(0)=0,D(q_nrf_v)(0)=0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 98 "req_q_nonRootedF ace_vertex_order2:=diffeqtorec(deq_q_nonRootedFace_vertex_order2,q_nrf _v(z),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "proc_q_nonRootedFace_vertex_order2:=rectoproc(req_q_n onRootedFace_vertex_order2,u(m),list):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 38 "proc_q_nonRootedFace_vertex_order 2(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 67 "dev_q_nonRootedFace_vertex_order2:=sum('%[i]*z**(i-1)','i'=1..N+1) :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 54 "We det ermine the development of q_vertex_vertex_order2" }{TEXT 223 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 164 "Eq_q_vertex_vertex_order2 _gamma:=2*(8*gamma^5+28*gamma^4+31*gamma^3+21*gamma^2+10*gamma+2)*gamm a^4+qvv2*((1+3*gamma+gamma^2)^2*(1+gamma)^2*(gamma-1)*(2*gamma+1)^3):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 97 "Eq_q_vertex_vertex_order2_z:=factor(resultant(Eq_q_v ertex_vertex_order2_gamma,Eq_gamma_z,gamma)):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 104 "deq_q_vertex_vertex_order 2:=algeqtodiffeq(Eq_q_vertex_vertex_order2_z,qvv2(z),\{qvv2(0)=0,D(qvv 2)(0)=0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 81 "req_q_vertex_vertex_order2:=diffeqtorec(deq_q_vertex_ vertex_order2,qvv2(z),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 " > " 0 "" {MPLTEXT 1 0 77 "proc_q_vertex_vertex_order2:=rectoproc(req_q _vertex_vertex_order2,u(m),list):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 31 "proc_q_vertex_vertex_order2(N):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 60 "dev_q _vertex_vertex_order2:=sum('%[i]*z**(i-1)','i'=1..N+1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 57 "We determine the deve lopment of q_vertex_vertex_orderNot2" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 106 "Eq_q_vertex_vertex_orderNot2_gamma:=2* gamma^2*(3*gamma+2)+qvv*((gamma-1)*(1+3*gamma+gamma^2)*(2*gamma+1)):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 103 "Eq_q_vertex_vertex_orderNot2_z:=factor(resultant(Eq_ q_vertex_vertex_orderNot2_gamma,Eq_gamma_z,gamma)):" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 95 "deq_q_vertex_vertex_o rderNot2:=algeqtodiffeq(Eq_q_vertex_vertex_orderNot2_z,qvv(z),\{qvv(0) =0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 86 "req_q_vertex_vertex_orderNot2:=diffeqtorec(deq_q_vertex_vertex_ord erNot2,qvv(z),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 75 "req_q_vertex_vertex_orderNot2:=subs(_C[0]=0,req_q_ve rtex_vertex_orderNot2):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 83 "proc_q_vertex_vertex_orderNot2:=rectoproc(req_q_v ertex_vertex_orderNot2,u(m),list):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 34 "proc_q_vertex_vertex_orderNot2(N) :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 63 "d ev_q_vertex_vertex_orderNot2:=sum('%[i]*z**(i-1)','i'=1..N+1):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 33 "We determi ne the development of q" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 113 "Eq_q_gamma:= gamma^6*(2*gamma^3-1-4*gamma-3*gamma^2 +gamma^4)+q*((1+gamma)^4*(1+3*gamma+gamma^2)^2*(2*gamma+1)^3):" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 55 "Eq_q_z:=factor(resultant(Eq_q_gamma,Eq_gamma_z,gamma) ):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 53 " deq_q:=algeqtodiffeq(Eq_q_z,q(z),\{q(0)=0,D(q)(0)=0\}):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 36 "req_q:=diffeqtor ec(deq_q,q(z),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 35 "proc_q:=rectoproc(req_q,u(m),list):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 10 "proc_q(N):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 39 "dev_q :=sum('%[i]*z**(i-1)','i'=1..N+1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 289 "> " 0 "" {MPLTEXT 1 235 278 "sum_k_rooted_3connected:=serie s(dev_q+z*subs(z=z**2,dev_q_rootedFace_vertex_order2+dev_q_nonRootedFa ce_vertex_order2)+z**2*subs(z=z**2,dev_q_rootedFace_nonRootedFace_orde r2+dev_q_nonRootedFace_nonRootedFace_order2)+subs(z=z**2,dev_q_vertex_ vertex_order2)+sum(phi(k)*subs(z=z**k," }{MPLTEXT 1 0 29 "dev_q_vertex _vertex_orderNot2" }{MPLTEXT 1 235 17 "),k=3..N),z,N+1):" }{MPLTEXT 1 200 0 "" }}}{EXCHG {PARA 288 "> " 0 "" {MPLTEXT 1 213 92 "non_rooted_3 connected_map:=series(1/2*int(series(sum_k_rooted_3connected/z,z,N+1), z),z,N+1);" }{MPLTEXT 1 213 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I :non_rooted_3connected_mapG6\"+_]lI\"zGF%\"\"\"\"\"'F(\"\")\"\"#\"\"* \"\"$\"#5\"\"%\"#6\"#:\"#7\"#K\"#8\"#*)\"#9\"$m#F1\"$(z\"#;\"%'\\#\"#< \"%7!)\"#=\"&Gg#\"#>\"&))e)\"#?\"'3mG\"#@\"';_'*\"#A\"(w(yK\"#B\")[:A6 \"#C\")#>l'Q\"#D\"*@00M\"\"#E\"*CAQn%\"#F\"+x-3Q;\"#G\"+[g))od\"#H\",J Ei2/#\"#I\",SyF%\\s\"#J\"-t`LF&e#F3\"-_y2A`#*\"#L\".GN0oKK$\"#M\"/M?4$ )Q(>\"\"#N\"/+-quLFV\"#O\"0cPppp$o:\"#P\"0w0?It'*p&\"#Q\"1y[X8rkw?\"#R \"1=3sxU`%e(\"#S\"2#>iWAX[wF\"#T\"3-C,_x-i=5\"#U\"3!y'R4l'[[u$\"#V\"4] ,Y8&3**[z8\"#W\"4))ois\"4=E\"4&\"#X\"5s5:z.ujU#)=\"#Y\"5w`2k7O#p?(p\"# Z\"6/x?Dt%[(ple#\"#[\"6%*yQE0sLZ6h*\"#\\\"73&*QWwh@@twN\"#]\"8#G(3NPPJ y-IL\"\"#^\"8#pssJ#Q.o'*[(\\\"#_\"9#Hgpt[(=+;=f=\"#`\"9gstH3t\\:-+dp\" #a\":q03_Io\"R>b`1E\"#b\":#)3S$*o8Gekxux*\"#c\";w:ik41C5$G;>n$\"#d\"<` OWddfhK)fI_!Q\"\"#e\"&\"#f\"=lKF0)GMlf`\"Rkd>\"#g\"=G`X O\">@[dX+\\JQ(\"#h\">`6!=3'pv,.c#[A(y#\"#i\"?u%4%fXLi1NS;0?`5\"#j\"?\" Q5$H[eH7]=k\\K$)R\"#k\"@cO#[B%=lr4pz#)oy]\"\"#l\"@)*e0z4C&RVrkDy%Gr&\" #m\"AWcx.\\5YQ6) >F?7$\"#p\"Cq2R$yq<;aqzat\"RH$>'Q\"\"#x\"GQmC*3j*zekK)[VY%zrw'H&\"#y\"H))ellOk Bn\"\\WbLHrZr8D?\"#z\"Ha<.sWM!o,IRr9tffjDZx\"#!)\"IY&[#*)\\['*R[+q?([w v57a'H\"#\")\"JK+>_4u!G#>jLHfnjn#**oN6\"##)\"J+lfUnad7u(*3/Su\"[XJx^V \"#$)\"K%ySSVyo2-$f$G?Hv**G#\\Ro;\"#%)\"K[U>(4iV_Yr_891T;sf!f*R'\"#&) \"L!*3V!fFLWi=sh,vT3cs%efX#\"#')\"L'otT55!H%*z&z/V\"G6h\"QEvH%*\"#()\" M+n0xie2er%H0[)>#4o7G1Bi$\"#))\"NkWs7%)3U/P&*H=#[j#eh]d,@R\"F5\"NwDPgb Q'z$4)o`DPbvs%f.bCN&\"#!*\"OmX\"[=VyYd(oti/JH*4mRHh)e?\"#\"*\"OU3]5Wqv /:n3#HD])\\!pif&)H#z\"##*\"PSc=#y!**fk?h.fQC%fD6_8g`6W)>uyu6\"#%*\"Qy.!ysR3r%RC8x,e3ib'*\\k?ys'\"#(* \"S!Gq(yTD')eA(fV'y8$f[^*RHm$f_f#\"#)*\"TUC-K%45YDga!o&=S\")zWQ\"*zh[ \\,5\"#**\"TSmaz\\`.!Hp#phJP>nBA^/D0C!oXF(=\"4OONIw/QZ-'p)\\\"$2\"\"YG7CtOa\\y( op6q>]oK0\"piXT0#)fGI>\"$3\"\"Y/=y7@$4$=r-N'[O>FthbPU'[PA#yPZ(\"$4\"\" ZcQymQy(3rUSr$3uz)4&f^![fZ_gOg%*G\"$5\"\"en=BfCZxTO\"\\R7_o8a_n#fYh/[t 4ST@6\"$6\"\"enU&))RCo3.%oN`rf` '\"$9\"\"gniHs&)*e=^oh/RBb.z`C=RAX\"4!R(RKULD\"$:\"\"gn[^Hm[s+wLHWu%z/ 3iT4e+/P\"ei=OeG)*\"$;\"\"hncuj4XB,P/%H&GGQReA53$36v')z3`P29Q\"$<\"\"i n#oVwnX\"*er>BnI\")='oi?E&eZ43$R]&4s/[\"\"$=\"\"in%fZ[zfa[&Q2et%Q!GiCY pD:8+'z7\"o]2[d\"$>\"\"jnu[EO/_8+yEd\")\\AI(*y#H4,CG&zh(e>lIKA\"$?\"\" jneo9+#=%*\\)>/gI!e/oL$=mH\\Lz!zA4PsYr')\"$@\"\"[o8#*)4;sw`Y8x!)z*3WK) z+m\\\"Hcg2kAT'y#pL\"$A\"\"\\o=*)>GN\"\\d$fu3Ox_fP>%yBW*)[(eFb-F_Q%48 \"$B\"\"\\o(*=fg7Dk5#Hje;;<\\.Bo+r[kfy4L@?q?!4&\"$C\"\"]oob4cXBxTkg-W' eO)oX9DT:k75U`.3aW=z>\"$D\"\"]oXmk*Q/))o96%>!H!4F]yCYP1R>\"3z\"4a(ycsp (\"$E\"\"^o!o8zIzbtrD*[x\"eU*oC\"e_yM/]@%*H\"$F\"\"_o*e3n7ugM8 *)pJ$Rle%*Q/I#)*HHbn7]))=&R&**\\;\"\"$G\"\"_o!)fyhGq,+.R`C*3yf>R$e8APc #ea7k)fxW!Q`%\"$H\"\"`og(3qbELWh>8b>p_!=X\\1fa!3(*znV`^*oH)yk<\"$I\"\" `o7/))**4dX@)z!yl')=qLj@=0z&>%ok()e)QB(z-4(o\"$J\"\"ao%eisT%o][\\&y3pE 9)[gg\"pDSi&Q;93>@)Q@Bcn#\"$K\"\"bo3S'y_R7'e#[CWD;WSP:#>o\")f')fS\"$M\"\"co%4 &fQ*y7[xCjYF;?/h_k=nPn[%*G]$)oLm[Y?>e\"\"$N\"\"coQCGo$[qnw(>sXma_9+^l> Uc`LG%>NnHc-KI^;'\"$O\"\"do+Skdd/n`ptD(*)*e(HD#)Rdk#\\g.)*pI<\"e\"\\PQ jJS#\"$P\"\"doG\\&ys'>t'o\"49MW,q%zKJ\\pQ&)znD;0;WQH!z'Gp$*\"$Q\"\"eo% o%ox_%)z<&HGA(o$R^Y:7=!)H'G!>mea@oC6Y%R^`O\"$R\"\"foY*=oFK5/@G?Zqm25(z R!4)[c#))G^8%)=3t)\\8HM\\U\"\"$S\"\"fo'RLi;s:cE:1J6T0Yb_SG/gU]zHfLEZ;g p(z*\\eb\"$T\"\"go;H0wqL$*=w/8D;Yx$>FO4E#*R&=XERi\\I H0YaSOAKrxN*G\"\"$X\"\"io7dULx!e94+4:%\\`9][t>Si`Bxx.!z6t5;?ih+,S.&\"$ Y\"\"joQ$\\?M>;MbdQHc#pFBdb3`b(3w#**QEU+OX[%fVJstl>\"$Z\"\"jo'pnLO`nV \"yQ#*Hb!yWnTOXt>N.8;/(pd6G^//*oxIxw\"$[\"\"[pG!R3<,jaeee.%>JV\"QZ$RDi 8*zVH\\<6#Gcc69P!f0*)*H\"$\\\"\"\\p;(z/b)3I^Q%Rh#*\\!\\l7QIroY\"[\"yi! e&p_P!p\\&[P-=;<\"\"$]\"-I\"OGI*protectedGF`]l6#F(\"$^\"" }{TEXT 20 0 "" }}}}}{SECT 1 {PARA 269 "" 0 "" {TEXT 207 57 "Enumeration according \+ to the number of vertices and faces" }{TEXT 207 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 238 "As the complexity of enumeration will be of O(N^2) operations (instead of O(N log(N)) for the fast enumeration ac cording to the number of edges, we have to lower a little the number N . However values of N till few hundreds seem acceptable" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 7 "N:=100;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I\"NG6\"\"$+\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 243 "We concentrate here on the case of 3-connected maps, which is more interesting. A similar tr eatment can be carried for 2-connected maps with the algebraic series \+ eta1(y1,y2) and eta2(y1,y2) instead of the series gamma1(z1,z2) and ga mma2(z1,z2)" }}}{SECT 1 {PARA 281 "" 0 "" {TEXT 218 40 "Enumeration of unrooted 3-connected maps" }{TEXT 218 0 "" }}{SECT 0 {PARA 284 "" 0 " " {TEXT -1 23 "Treatment on an example" }{TEXT -1 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 153 "To perform the treatment of this part, you \+ have either to have a recent version of maple (at least version 9) and use the command with(gfun) as follows. " }{TEXT 223 0 "" }{TEXT 223 513 "\nYou can also download algolib for your version of maple at the \+ following adress http://algo.inria.fr/libraries/. Then, before the ins truction with(gfun); you add the instruction libname:=\"PATH\"; where \+ PATH is the path of the repertory where you have installed algolib (se e README at the web adress). If you do not want to install algolib, yo u can just read this section without executing and pass directly to th e next section where the development of all series of k-rooted irreduc ible quadrangulations are given" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 " > " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#7PI(LaplaceG6\"I.algebraicsubsGF%I.algeqtodiffe qGF%I.algeqtoseriesGF%I.algfuntoalgeqGF%I&borelGF%I.cauchyproductGF%I. diffeq*diffeqGF%I.diffeq+diffeqGF%I,diffeqtableGF%I2diffeqtohomdiffeqG F%I,diffeqtorecGF%I)guesseqnGF%I(guessgfGF%I0hadamardproductGF%I0holex prtodiffeqGF%I)invborelGF%I,listtoalgeqGF%I-listtodiffeqGF%I0listtohyp ergeomGF%I+listtolistGF%I.listtoratpolyGF%I*listtorecGF%I-listtoseries GF%I,maxdegcoeffGF%I*maxdegeqnGF%I,maxordereqnGF%I,mindegcoeffGF%I*min degeqnGF%I,minordereqnGF%I*optionsgfGF%I,poltodiffeqGF%I)poltorecGF%I/ ratpolytocoeffGF%I(rec*recGF%I(rec+recGF%I,rectodiffeqGF%I,rectohomrec GF%I*rectoprocGF%I.seriestoalgeqGF%I/seriestodiffeqGF%I2seriestohyperg eomGF%I-seriestolistGF%I0seriestoratpolyGF%I,seriestorecGF%I/seriestos eriesGF%" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 410 "We take the example of the series q_rooted_blackvertex, the series of 2- rooted irreducible quadrangulations of type face-black vertex whose 2- root is incident to the axial face. We shortly write qrb for this ser ies. For this series, the calculations are not too long (for other ser ies of k-rooted irreducible quadrangulations, the steps of the calcula tion are the same but can take more time, up to 2 hours). " }{TEXT 223 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 71 "First we give again the two equations linking z1, z2, gamma1 and gamma2" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 46 "Eq_z1_gamma1_gamma2:=-z 1*(1+gamma2)**2+gamma1;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I4Eq_z1_gamma1_gamma2G6\",&*&I#z1GF%\"\"\",&F)F)I'gamm a2GF%F)\"\"#!\"\"I'gamma1GF%F)" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> \+ " 0 "" {MPLTEXT 1 0 46 "Eq_z2_gamma1_gamma2:=-z2*(1+gamma1)**2+gamma2; " }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I4Eq_z2_gamma 1_gamma2G6\",&*&I#z2GF%\"\"\",&F)F)I'gamma1GF%F)\"\"#!\"\"I'gamma2GF%F )" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 209 "For the c alculations we will also need an equation linking only z1, z2 and gamm a1. Such an equation is easily obtained by taking the resultant of Eq_ z1_gamma1_gamma2 and Eq_z2_gamma1_gamma2 according to gamma2" }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 83 "Eq_z1_z2_gamma 1:=factor(resultant(Eq_z1_gamma1_gamma2,Eq_z2_gamma1_gamma2,gamma2));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I0Eq_z1_z2_gam ma1G6\",6I#z1GF%!\"\"I'gamma1GF%\"\"\"*&F'F*I#z2GF%F*!\"#*&F'F*F,\"\"# F(*(F'F*F,F/F)F*!\"%*(F'F*F,F/F)F/!\"'*(F'F*F,F*F)F*F1*(F,F/F)\"\"$F'F *F1*(F'F*F,F*F)F/F-*(F,F/F)\"\"%F'F*F(" }{TEXT 20 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 177 "From the (gamma1,gamma2)-rational exp ression of qrb that we have obtained precedently, we get an algebraic \+ equation (just by taking the numerator) between qrb, gamma1 and gamma2 " }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 113 "Eq_q _rooted_blackVertex_gamma1_gamma2:=-qrb*((1+2*gamma2+gamma2^2+gamma1)* (gamma1+1+gamma2)^2)+gamma2^2*gamma1^2:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 396 "As we want the development of qrb in \+ z1 and z2, what we want is an algebraic equation linking qrb, z1 and z 2. To get this equation, we first take the resultant of Eq_qrb_gamma1_ gamma2 and Eq_z2_gamma1_gamma2 to get rid of gamma2. Then we take the \+ resultant of this equation and of Eq_z1_z2_gamma1 to get rid of gamma1 . We finally obtain an albebraic equation Eq_qrb_z1_z2 linking qrb, z1 and z2. " }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 119 "Eq_q_rooted_blackVertex_z2_gamma1:=factor(resultant(Eq_q_rooted_b lackVertex_gamma1_gamma2,Eq_z2_gamma1_gamma2,gamma2)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 107 "Eq_q_rooted_black Vertex_z1_z2:=factor(resultant(Eq_q_rooted_blackVertex_z2_gamma1,Eq_z1 _z2_gamma1,gamma1)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 844 "Here we intrduce a calculation trick which will allows \+ us to work only with an algebraic equation in one variable t. Instead \+ of considering (z1,z2)->qrb(z1,z2), we consider the series in one vari able t-> qrb(t*z1,t*z2). We denote this series by qrb(t). This series \+ has one variable t and the coefficients of t^n is not a rational numbe r but is a polynomial in the two indeterminates z1 and z2 (this polyno mial is homogenous of degree n). As a consequence, the field of coeffi cients is not Q anymore, but Q[z1,z2], which will make no difference f or the calculations (a product of two elements of the field will just be a product of polynomials of z1 and z2 instead of being a product o f two rational numbers). The series qrb(t) is algebraic and its algebr aic equation is just obtained by substituting z1 and z2 by t*z1 and t* z2 in Eq_qrb_z1_z2." }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 766 "Eq_q_rooted_blackVertex_t_z1_z2:=subs(\{z1=z1*t,z2=z 2*t\},12*z2*z1^2*qrb^2+6*z1^3*qrb*z2+18*z1^3*z2^2*qrb^2+6*z1^2*z2^2*qr b^2+z1^5*z2^2+z1^4*z2^2-z1^2*qrb+26*z1^3*qrb^2*z2+3*z1^3*qrb^2+4*z1^5* z2^2*qrb+3*z1^3*qrb^4+8*z1^2*z2*qrb^4+4*z1^3*z2^2*qrb+z1^5*z2^2*qrb^4+ 5*z1^4*z2^2*qrb^4+10*z1^3*z2^2*qrb^4+30*z1^3*z2*qrb^3-6*z1^3*qrb^3+16* z1^2*z2^2*qrb^3+22*z1^2*z2*qrb^3-3*z1^2*qrb^3+3*z1^2*qrb^2+4*z1^5*z2^2 *qrb^3+z2^2*qrb^4+5*z1*z2^2*qrb^4+z1^2*qrb^4+2*z1*z2*qrb^4+6*z1^5*z2^2 *qrb^2+2*z1^5*z2*qrb^4+8*z1^4*z2*qrb^4+12*z1^3*z2*qrb^4+10*z1^2*z2^2*q rb^4+4*z1^5*z2*qrb^3+4*z1*z2^2*qrb^3+6*z1*z2*qrb^3+2*z1^5*z2*qrb^2+z1^ 5*qrb^4+3*z1^4*qrb^4+8*z1^4*z2^2*qrb+16*z1^4*z2^2*qrb^3+18*z1^4*z2^2*q rb^2+24*z1^3*z2^2*qrb^3+18*z1^4*z2*qrb^3+16*z1^4*z2*qrb^2-3*z1^4*qrb^3 +6*z1^4*z2*qrb):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 290 "" 0 "" {TEXT -1 85 "We perform the same treatment as for the enumeration acco rding to the number of edges" }{TEXT -1 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "deq_q_rooted_blackVertex:=algeqtodiffeq(Eq_q_roo ted_blackVertex_t_z1_z2,qrb(t),\{qrb(0)=0\}):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 76 "rec_q_rooted_blackVertex:= diffeqtorec(deq_q_rooted_blackVertex,qrb(t),u(m)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 73 "proc_q_rooted_blackVer tex:=rectoproc(rec_q_rooted_blackVertex,u(m),list):" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 276 "" 0 "" {TEXT 223 168 "As rectoproc deals not au tomatically with polynomials, we have to add an instruction forcing th e procedure to simplify the expression obtained for the coefficient t^ n. " }{TEXT 223 0 "" }}{PARA 276 "" 0 "" {TEXT 223 165 "Indeed, the co efficient of t^n obtained from the recurrence is at first sight a rati onal expression of z1 and z2. But it turns out that this expression ca n always be" }{TEXT 223 0 "" }}{PARA 276 "" 0 "" {TEXT 223 169 " simpl ified to a polynomial, which can be done with the instruction factor, \+ followed by the instruction expand to have a standard expression for t he obtained polynomial." }{TEXT 223 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 81 "proc_q_rooted_blackVertex := proc (n) local i, u; u[ 0] := 0; u[1] := 0; u[2] := \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n0; u[3] := 0; u[4] := z1^2*z2^2; u[5] := z1^3*z2^2; u[6] := 6*z1^3*z 2^3; u[7] :=\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n 12*z1^4*z2^3+4 *z1^3*z2^4; u[8] := 6*z1^5*z2^3+51*z1^4*z2^4; u[9] := 129*z1^5*z2\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^4+60*z1^4*z2^5; u[10] := 121*z1 ^6*z2^4+512*z1^5*z2^5+22*z1^4*z2^6; u[11] := 39*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^7*z2^4+1448*z2^5*z1^6+794*z1^5*z2^6; u[12] := \+ 1872*z1^7*z2^5+5635*z1^6*z2^6+5\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n46*z1^5*z2^7; u[13] := 1148*z1^8*z2^5+17035*z1^7*z2^6+10360*z1^6 *z2^7+140*z1^5*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 96 "\n2^8; for i \+ from 14 to n do u[i] := expand(factor(-(-6104678400*z1^10*z2^7*u[i-13] +(-45719856*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^2*z2^5+66721 536*z1^4*z2^3+68081472*z1^5*z2^2+393120*z1*z2^6-20078064*z1^3*z2^4\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+19279620*z1^6*z2)*u[i-3]+(8754 614400*z1^4*z2^6-3030754560*z1^5*z2^5-13406038368\\" }{MPLTEXT 1 0 0 " " }{MPLTEXT 1 0 82 "\n*z1^6*z2^4-767784240*z1^8*z2^2-5918193600*z1^7*z 2^3+1437004800*z2^7*z1^3)*u[i-6]\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+(46197895680*z1^6*z2^7-27668883552*z1^7*z2^6-45643877280*z1^9*z 2^4-134303354880\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^8*z2^5- 1020418560*z1^10*z2^3)*u[i-9]+(-22078586880*z1^10*z2^6-18882109440*z1 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^9*z2^7)*u[i-12]+(6422490*z 1^5*z2+3257496*z1^2*z2^4-2599776*z1*z2^5+4503924*z1^3\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z2^3+5050188*z1^4*z2^2)*u[i-2]+(-946514016 *z1^4*z2^5-1681801704*z1^5*z2^4+11572\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n10208*z2^6*z1^3-778186152*z1^6*z2^3-461982240*z1^7*z2^2+1001 54880*z2^7*z1^2)*u[i\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-5]+(30 692874240*z1^5*z2^7+28485921600*z1^6*z2^6-86708917008*z1^7*z2^5-662143 714\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n08*z1^8*z2^4-5737127040* z1^9*z2^3)*u[i-8]+(-10812787200*z1^8*z2^7-76177059840*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^9*z2^6-26704995840*z1^10*z2^5)*u[i-11]+( 133650*z1^4*z2-9720*z1*z2^4+144828*z1^2\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n*z2^3+231336*z1^3*z2^2)*u[i-1]+(-168547032*z1^4*z2^ 4+84813120*z1^2*z2^6+12990780\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n*z1^7*z2-420512400*z1^3*z2^5+335963088*z1^5*z2^3+38390220*z1^6*z2^2) *u[i-4]+(969\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0831360*z2^7*z1 ^4-330674400*z1^9*z2^2-44191438464*z1^7*z2^4-25179250032*z2^5*z1^\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6-9583115280*z1^8*z2^3+274307113 92*z1^5*z2^6)*u[i-7]+(-11751505920*z1^10*z2^4+25\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n823646720*z1^7*z2^7-97017050880*z1^9*z2^5-9050439 0912*z1^8*z2^6)*u[i-10]+((34535\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n11680*z1^10*z2^4+27194616320*z1^8*z2^6+29511332352*z1^9*z2^5-933 1015680*z1^7*z2^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7)*u[i-10]+ 1527939072*z1^10*z2^7*u[i-13]+(-16528860*z1^6*z2-502848*z1*z2^6-49912 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n848*z1^5*z2^2+10345104*z1^3 *z2^4-42471486*z1^4*z2^3+31838400*z1^2*z2^5)*u[i-3]+(\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-3953219584*z1^4*z2^6+2944937996*z1^7*z2^3+ 1560944208*z1^5*z2^5+6180238668*z1^6*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz2^4-710344704*z2^7*z1^3+403178544*z1^8*z2^2)*u[i-6]+(148655 85984*z1^9*z2^4-1644\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0775680 *z1^6*z2^7+334347264*z1^10*z2^3+44706643968*z1^8*z2^5+8288244032*z1^7* z2^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6)*u[i-9]+(4786679808*z1 ^9*z2^7+5875200000*z1^10*z2^6)*u[i-12]+(-4457430*z1^3*z2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^3-4911354*z1^4*z2^2-3903228*z1^2*z2^4-68 52357*z1^5*z2+2450736*z1*z2^5)*u[i-2]+(\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n549022848*z1^4*z2^5-619885632*z2^6*z1^3+530033736*z 1^6*z2^3+278387460*z1^7*z2^2+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n929331228*z1^5*z2^4-58752000*z2^7*z1^2)*u[i-5]+(2190337920*z1^9*z2^3 +31870072752\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^7*z2^5-1094 7938304*z1^6*z2^6+23945254320*z1^8*z2^4-11672875008*z1^5*z2^7)*u[\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\ni-8]+(21578840064*z1^9*z2^6+2143 558656*z1^8*z2^7+7466425344*z1^10*z2^5)*u[i-11]+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n(-201690*z1^2*z2^3+13122*z1^5-233523*z1^4*z2-4500 36*z1^3*z2^2+30132*z1*z2^4)*u[i\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-1]+(-9893988*z1^7*z2+249263568*z1^3*z2^5+115288920*z1^4*z2^4-14 169546*z1^6*z2^2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-167067306* z1^5*z2^3-56166912*z1^2*z2^6)*u[i-4]+(-11134204096*z1^5*z2^6-40995256 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n32*z2^7*z1^4+17885103744*z1 ^7*z2^4+10440133056*z2^5*z1^6+4142069600*z1^8*z2^3+15\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6064320*z1^9*z2^2)*u[i-7]+((-599292096*z1^8 *z2^3+1499893248*z1^5*z2^6-2426357568\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^7*z2^4+580116480*z2^7*z1^4-1446093504*z2^5*z1^6-24261120 *z1^9*z2^2)*u[i-7]+(\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-299953 1520*z1^9*z2^5-338411520*z1^10*z2^4-2739443712*z1^8*z2^6+1099530240*z1 ^7*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz2^7)*u[i-10]-127401984* z1^10*z2^7*u[i-13]+(-6630336*z1^2*z2^5+207360*z1*z2^6+46\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n45674*z1^6*z2+7424946*z1^4*z2^3-748440*z1 ^3*z2^4+11485800*z1^5*z2^2)*u[i-3]+(593\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n983488*z1^4*z2^6+117080064*z2^7*z1^3-956011896*z1^6 *z2^4-266160960*z1^5*z2^5-701\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n63136*z1^8*z2^2-491551512*z1^7*z2^3)*u[i-6]+(1948262400*z1^6*z2^7-36 495360*z1^10\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z2^3-497432678 4*z1^8*z2^5-836860416*z1^7*z2^6-1619735040*z1^9*z2^4)*u[i-9]+(-52\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1994240*z1^10*z2^6-404029440*z1^ 9*z2^7)*u[i-12]+(1308798*z1^3*z2^3+1535760*z1^2*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz2^4+1438074*z1^4*z2^2+2303154*z1^5*z2-668736*z1* z2^5)*u[i-2]+(111002112*z2^6*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^3-55851804*z1^7*z2^2-119092968*z1^6*z2^3-173899404*z1^5*z2^4-10 2835584*z1^4*z2^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n5+11427840* z2^7*z1^2)*u[i-5]+(-279309312*z1^9*z2^3+1384611840*z1^6*z2^6-29018864 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n64*z1^8*z2^4-3913376256*z1^ 7*z2^5+1483800576*z1^5*z2^7)*u[i-8]+(-696508416*z1^10\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z2^5-2044624896*z1^9*z2^6-116023296*z1^8*z 2^7)*u[i-11]+(116397*z1^4*z2-19683*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^5+270216*z1^3*z2^2-28188*z1*z2^4+49086*z1^2*z2^3)*u[i-1]+(2 2185000*z1^5*z2^3+24\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n34860*z 1^7*z2+12460032*z1^2*z2^6-46657152*z1^3*z2^5-25985772*z1^4*z2^4+355032 *z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^6*z2^2)*u[i-4]+((-93081 6*z1^2*z2^6-350262*z1^5*z2^3+201438*z1^6*z2^2-193428*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7*z2+2657088*z1^3*z2^5+1916460*z1^4*z2^4) *u[i-4]+(29033344*z1^8*z2^3-67011584*z1\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n^5*z2^6-27475968*z2^7*z1^4+66965760*z2^5*z1^6+12441 60*z1^9*z2^2+110477856*z1^7*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n2^4)*u[i-7]+(11059200*z1^10*z2^4+101864448*z1^9*z2^5+92578816*z1^8*z 2^6-42516480\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^7*z2^7)*u[i -10]+3538944*z1^10*z2^7*u[i-13]+(-27648*z1*z2^6+366336*z1^2*z2^5-\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n802764*z1^5*z2^2-426222*z1^6*z2- 157248*z1^3*z2^4-227448*z1^4*z2^3)*u[i-3]+(15031\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n488*z1^5*z2^5-29717504*z1^4*z2^6+4047552*z1^8*z2^ 2-6438912*z2^7*z1^3+49730316*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^6*z2^4+27544348*z1^7*z2^3)*u[i-6]+(28687360*z1^7*z2^6+1327104*z 1^10*z2^3+590653\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n44*z1^9*z2^ 4+185037312*z1^8*z2^5-76922880*z1^6*z2^7)*u[i-9]+(15482880*z1^10*z2^6 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+11354112*z1^9*z2^7)*u[i-12 ]+(-103032*z1^3*z2^3-199260*z1^2*z2^4-241299*z1^5*z2-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n122472*z1^4*z2^2+46656*z1*z2^5)*u[i-2]+(-66 63168*z2^6*z1^3+3732264*z1^7*z2^2+886\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1256*z1^6*z2^3-737280*z2^7*z1^2+6170112*z1^4*z2^5+11051712*z 1^5*z2^4)*u[i-5]+(-5\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7621504 *z1^6*z2^6+117935808*z1^8*z2^4+11897856*z1^9*z2^3-63062016*z1^5*z2^7+1 606\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n15872*z1^7*z2^5)*u[i-8]+ (64806912*z1^9*z2^6+21676032*z1^10*z2^5+933888*z1^8*z2^7\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n)*u[i-11]+(6561*z1^5-51516*z1^3*z2^2-1652 4*z1^4*z2+7776*z1*z2^4+7776*z1^2*z2^3)*\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\nu[i-1])*i)*i)*i)/((-8748*z1^2*z2^2+26244*z1*z2^3+58 32*z2^4-16038*z1^3*z2+13122*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n1^4+(24057*z1^3*z2-39366*z1*z2^3+13122*z1^2*z2^2-8748*z2^4-19683*z1^ 4+(2916*z2^4\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 84 "\n-4374*z1^2*z2^2 +13122*z1*z2^3+6561*z1^4-8019*z1^3*z2)*i)*i)*i))) end do; [seq(u[i]\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 24 "\n, i = 0 .. n)] end proc" } {MPLTEXT 1 0 0 "" }}{PARA 272 "" 1 "" {TEXT 230 74 "Warning, inserted \+ missing semicolon at end of statement, ... n)] end proc;" }{TEXT 230 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I:proc_q_rooted_blackVertexG6 \"f*6#I\"nGF%6$I\"iGF%I\"uGF%F%F%C2>&8%6#\"\"!F1>&F/6#\"\"\"F1>&F/6#\" \"#F1>&F/6#\"\"$F1>&F/6#\"\"%*&I#z1GF%F9I#z2GF%F9>&F/6#\"\"&*&FCF=FDF9 >&F/6#\"\"',$*&FCF=FDF=FM>&F/6#\"\"(,&*&FCF=FDFAFA*&FCFAFDF=\"#7>&F/6# 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\"L!e&oH!*\\%=H'3jOV[!>)zi`Mv(*(F(FiwF*FUF,Figm\"I0b@@3;-`FOV&4>wn'4$* zn*(F(FbxF*FXF,F\\\\m\"IW:oay&\\AkQ8!\\> %HY*Hb#f>?;\"HA(owX!\\-]-=*(F(FbxF*F_tF,F-\"X+&fOCO3y4yV,e$RZ;A(Rgr-_C ^A>7*(F(F7F*FPF,F\\\\m\"FHH,8/**fA8/FEn9e$)>Y'*(F(FYF*FexF,F\\\\m\"J/W $f-s#3@*>wX5>;E35(*)y&*(F(FXF*FbxF,F\\\\m\"I#e0UY1CY*)4UoG$Q[rHJ(*H*(F *FLF(FdyF,Fiil\"F*FFF,F\\\\m\"N5`1HC'p3gZ!RG7D: Z`br7%\\9*(F(F`wF*FEF,F\\\\m\"?'\\Q^9J-p!\\6YyQ9>*(F(F\\pF*FQF,F\\\\m \"L/Nf_?#eu%y^=,k#p)oJY+Zn)*(F(FhoF*FMF,F\\\\m\"M?/r)4K*H')**>eCAgqu%* p*GAO$*(F(FPF*FexF,Fip\"F+^6(3!)z*y#[/q:`m'pjB&" }{TEXT 20 0 "" }}}} {SECT 0 {PARA 284 "" 0 "" {TEXT -1 80 "The development of all series o f k-rooted irreducible quadrangulations are given" }{TEXT -1 0 "" }} {EXCHG {PARA 276 "" 0 "" {TEXT 223 66 "We advise to use a recent versi on of maple. For example version 9." }}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 46 "The series q_blackVertex_blackVertex_orderNot2" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 91 "proc_qbb_orderNo t2 := proc (n) local i, u; u[0] := 0; u[1] := 0; u[2] := 0; u[3] := 3* z1; \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nu[4] := 4*z1^2; u[5] := 25*z1^2; u[6] := 54*z1^3+21*z1^2; u[7] := 28*z1^4+273*z1\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^3; u[8] := 708*z1^4+384*z1^3; u [9] := 675*z1^5+3321*z1^4+165*z1^3; u[10] := 220\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^6+9500*z1^5+5995*z1^4; u[11] := 12386*z1^6+42 878*z1^5+4719*z1^4; u[12] := 76\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n44*z1^7+130450*z1^6+90012*z1^5+1365*z1^4; u[13] := 1820*z1^8+207 025*z1^7+575107*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 96 "\nz1^6+97643* z1^5; for i from 14 to n do u[i] := expand(factor(-(-32558284800*z1^8* u[i-13]+(-100\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n440000*z1^3+31 449600*z1^7-16135200*z1^2+3764946528*z1^5+1542715200*z1^6+21746171\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n52*z1^4+129149640*z1)*u[i-3]+(- 28601642592*z1^3-20404224000*z1^8-154701511680*z1\\" }{MPLTEXT 1 0 0 " " }{MPLTEXT 1 0 82 "\n^7-2171292480*z1^2-165845059104*z1^5-26140872460 8*z1^6-74047178784*z1^4)*u[i-6]+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n(-846289896960*z1^6-1204807853568*z1^7+18468311040*z1^4-41843360 5632*z1^8-693846\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n37440*z1^5) *u[i-9]+(-161190641664*z1^8-94826004480*z1^7)*u[i-12]+(144322560*z1^6 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+531121212*z1^3+484092936*z 1^4+467127648*z1^5+580074048*z1^2+204725070*z1)*u[i-2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n]+(-30767340240*z1^3-1602478080*z1^8-266950 51776*z1^7-5579942040*z1^2-3379077676\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n8*z1^5-48825108864*z1^6-38573633664*z1^4)*u[i-5]+(-983486039 616*z1^6+23995388736\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^4-2 53826334720*z1^8-946213466112*z1^7+2112721920*z1^3-236780859648*z1^5)* u[i\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-8]+(-459840356352*z1^7- 77077094400*z1^6-355222609920*z1^8)*u[i-11]+(-933120*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n5+126703116*z1^2+65968668*z1+151632*z1^4+ 61574256*z1^3+590490)*u[i-1]+(-10861019\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n820*z1^3-1570867200*z1^7-3883325760*z1^2+5801785920 *z1^5+425757888*z1^6-35095758\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n48*z1^4-13267800*z1)*u[i-4]+(-668888012736*z1^6-295400789664*z1^5-97 779916800*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^8-484298936832 *z1^7-6011458560*z1^3-35149063584*z1^4)*u[i-7]+(-395212713984*z1^\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6+1546214400*z1^5-978298899456*z 1^7-470175694848*z1^8)*u[i-10]+((299985283072*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^7+117328134144*z1^6-1789175808*z1^5+147060228096 *z1^8)*u[i-10]+8149008384*z1^8*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nu[i-13]+(-40227840*z1^7+770748264*z1^3-847033848*z1^4+330705666* z1^2-76603320*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-2205960480 *z1^5-961110144*z1^6)*u[i-3]+(84967537056*z1^5+72922322944*z1^7+43844 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n124256*z1^4+10169942016*z1^ 8+123887833344*z1^6+16831170048*z1^3+1321142400*z1^2)\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*u[i-6]+(405767627776*z1^7+21941720064*z1^5 +279215953920*z1^6+145933418496*z1^8-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6320664576*z1^4)*u[i-9]+(24891162624*z1^7+42654302208*z1^8)* u[i-12]+(-353994948*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^3-36 0202464*z1^5-427088538*z1^2-119024640*z1^6-158431383*z1-332883432*z1^4 )*u\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[i-2]+(23487170496*z1^5+ 14133292032*z1^7+26635915680*z1^4+940032000*z1^8+2787632\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7168*z1^6+19598720616*z1^3+3652245072*z1^ 2)*u[i-5]+(99201908736*z1^8-812574720*z\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n1^3-5773541760*z1^4+90618798336*z1^5+361297695360*z 1^6+353985669120*z1^7)*u[i-8]\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n+(101150834688*z1^8+20461068288*z1^6+129841569792*z1^7)*u[i-11]+(-10 82565+247471\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2*z1^4-66230136 *z1^3-137519046*z1^2+3203712*z1^5-71568846*z1)*u[i-1]+(893048832*\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^7+8566219188*z1^3+4936621536* z1^4+2960856288*z1^2+16971120*z1-1222985088*z1^5\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+521599104*z1^6)*u[i-4]+(42913087488*z1^8+3925463 808*z1^3+23207926816*z1^4+12899\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0404320*z1^5+276323998080*z1^6+202975224832*z1^7)*u[i-7]+((-4485 545472*z1^4-3813\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0066432*z1^ 6-773019648*z1^3-6285557760*z1^8-28466061312*z1^7-18797803008*z1^5)*u \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[i-7]+(319832064*z1^5-30781 784064*z1^7-11616288768*z1^6-15388901376*z1^8)*u[i-10\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n]-679477248*z1^8*u[i-13]+(435041280*z1^5+19 530720*z1^4+10555920*z1+16588800*z1^7\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+217230336*z1^6-391874112*z1^3-177571764*z1^2)*u[i-3]+(-1951 9285248*z1^6-1148244\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7872*z1 ^7-14374005120*z1^5-8409507456*z1^4-3261000960*z1^3-1689255936*z1^8-26 141\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1840*z1^2)*u[i-6]+(-3079 1725056*z1^6-45753507840*z1^7+721059840*z1^4-17025859584\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^8-2329325568*z1^5)*u[i-9]+(-376543641 6*z1^8-2179989504*z1^7)*u[i-12]+(686854\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n08*z1^4+34836480*z1^6+62472384*z1^3+94338432*z1^5+9 4401612*z1^2+38717190*z1)*u[i\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n-2]+(-5174728704*z1^6-2479251456*z1^7-5038933248*z1^5-5921584272*z1^ 4-4202649648\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^3-182845440 *z1^8-795056256*z1^2)*u[i-5]+(296045568*z1^4-44379823104*z1^6+103\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n956480*z1^3-12954107904*z1^8-443 38397184*z1^7-11625418752*z1^5)*u[i-8]+(-9623961\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n600*z1^8-12255363072*z1^7-1797783552*z1^6)*u[i-11 ]+(24823908*z1-2142288*z1^4-217\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7280*z1^5+590490+22694256*z1^3+47728116*z1^2)*u[i-1]+(-146544768 *z1^5-1621560528\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^4-69984 00*z1-154976256*z1^7-196512768*z1^6-2251431792*z1^3-763059744*z1^2)*u \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[i-4]+((13151232*z1^6+31029 696*z1^5+66256272*z1^2+933120*z1+157329144*z1^4+19756\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0152*z1^3+7225344*z1^7)*u[i-4]+(1756723200* z1^6+307494912*z1^8+914948736*z1^5+27\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1206272*z1^4+1335836672*z1^7+47929344*z1^3)*u[i-7]+(38333644 8*z1^6+1056751616*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^7-1563 0336*z1^5+538705920*z1^8)*u[i-10]+18874368*z1^8*u[i-13]+(-18855936*z1^ 6-3\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0188160*z1^5+24407730*z1 ^2+204120*z1+14468112*z1^4+52208496*z1^3-2211840*z1^7)*u\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[i-3]+(93585408*z1^8+1021486080*z1^6+8056 32960*z1^5+209144832*z1^3+529540544*z1^\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n4+16865280*z1^2+603471872*z1^7)*u[i-6]+(1134557184* z1^6+664535040*z1^8-27426816*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \nz1^4+1726889984*z1^7+83109888*z1^5)*u[i-9]+(110886912*z1^8+63700992* z1^7)*u[i-12\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n]+(-3732480*z1^ 6-8724672*z1^5-5497146*z1^2-2884653*z1-3837456*z1^4-1162512*z1^3)\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*u[i-2]+(11796480*z1^8+311740416 *z1^6+344145024*z1^5+303274872*z1^3+431322120*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^4+57677184*z1^2+143179776*z1^7)*u[i-5]+(18221230 08*z1^6+565444608*z1^8+49987891\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2*z1^5+6580224*z1^4+1859174400*z1^7-4423680*z1^3)*u[i-8]+(386629 632*z1^7+3059220\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n48*z1^8+521 99424*z1^6)*u[i-11]+(-5322186*z1^2+404352*z1^5-98415+433512*z1^4-2486 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n376*z1^3-2765826*z1)*u[i-1] )*i)*i)*i)/(590490+1154736*z1^3+5170068*z1^2-1399680*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^5-2344464*z1^4+3661038*z1+(-1082565+4298 184*z1^4-2117016*z1^3-9478458*z1^2+25\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n66080*z1^5-6711903*z1+(590490+1154736*z1^3+5170068*z1^2-1399 680*z1^5-2344464*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4+36610 38*z1+(233280*z1^5+390744*z1^4-192456*z1^3-861678*z1^2-610173*z1-98415 )*i\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 52 "\n)*i)*i))) end do; [seq(u [i], i = 0 .. n)] end proc;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3proc_qbb_orderNot2G6\"f*6#I\"nGF%6$I\"iGF%I\"uGF%F%F %C2>&8%6#\"\"!F1>&F/6#\"\"\"F1>&F/6#\"\"#F1>&F/6#\"\"$,$I#z1GF%F=>&F/6 #\"\"%,$*$F?F9FC>&F/6#\"\"&,$FE\"#D>&F/6#\"\"',&*$F?F=\"#aFE\"#@>&F/6# \"\"(,&*$F?FC\"#GFQ\"$t#>&F/6#\"\"),&FY\"$3(FQ\"$%Q>&F/6#\"\"*,(*$F?FI \"$v'FY\"%@LFQ\"$l\">&F/6#\"#5,(*$F?FO\"$?#Fbo\"%+&*FY\"%&*f>&F/6#\"#6 ,(F[p\"&'Q7Fbo\"&yG%FY\"%>Z>&F/6#\"#7,**$F?FW\"%WwF[p\"']/8Fbo\"&7+*FY \"%l8>&F/6#\"#8,**$F?Fin\"%?=F\\q\"'Dq?F[p\"'2^dFbo\"&Vw*?(8$\"#9F59$I %trueGI*protectedGF`r>&F/6#F\\r-I'expandGF`r6#-I'factorGI(_syslibGF%6# ,$*&,>*&F?Fin&F/6#,&F\\rF5!#8F5F5!,+[GeD$*&,0FQ!*++W+\"F\\q\")+'\\9$FE !)+_8;Fbo\"+Gl%\\w$F[p\"++_rU:FY\"+_rhu@F?\"*S'\\\"H\"F5&F/6#,&F\\rF5! \"$F5F5F5*&,0FQ!,#fU;gGFfq!,+SA//#F\\q!-!o6:qa\"FE!+![#Hr@Fbo!-/\"f]%e ;F[p!-3Ys39EFY!,%yyr/uF5&F/6#,&F\\rF5!\"'F5F5F5*&,,F[p!-gp*)*GY)F\\q!. oN&y![?\"FY\",S5Jo%=Ffq!-KcgL%=%Fbo!,Suj%QpF5&F/6#,&F\\rF5!\"*F5F5F5*& ,&Ffq!-k;k!>h\"F\\q!,![/g#[*F5&F/6#,&F\\rF5!#7F5F5F5*&,.F[p\"*gDKW\"FQ \"*777J&FY\"*OH4%[Fbo\"*[w7n%FE\"*[S2!eF?\"*q]s/#F5&F/6#,&F\\rF5!\"#F5 F5F5*&,0FQ!,S-Mn2$Ffq!+!3yCg\"F\\q!,w<0&pEFE!+S?%*zbFbo!,onx!zLF[p!,k) 3^#)[FY!,kOjt&QF5&F/6#,&F\\rF5!\"&F5F5F5*&,.F[p!-;'Rg[$)*FY\",O()Q&*R# Ffq!-?ZLEQDF\\q!-7hY8i%*FQ\"+?>s7@Fbo!-['f3yO#F5&F/6#,&F\\rF5!\")F5F5F 5*&,(F\\q!-_jNS)f%F[p!,+W4xq(Ffq!-?*4EAb$F5&F/6#,&F\\rF5!#6F5F5F5*&,.F bo!'?J$*FE\"*;JqE\"F?\")o'of'FY\"'K;:FQ\")cUdh\"'!\\!fF5F5&F/6#,&F\\rF 5!\"\"F5F5F5*&,0FQ!,?)>5'3\"F\\q!++s'3d\"FE!+gdK$)QFbo\"+?fy,eF[p\"*)) yvD%FY!+[ed4NF?!)+yE8F5&F/6#,&F\\rF5!\"%F5F5F5*&,.F[p!-OF,)))o'Fbo!-k' *y+aHFfq!,+o\"*zx*F\\q!-Ko$*)H%[FQ!+g&e9,'FY!,%ej!\\^$F5&F/6#,&F\\rF5! \"(F5F5F5*&,*F[p!-%)Rr7_RFbo\"++W@Y:F\\q!-c%**))Hy*Ffq!-[[pv,ZF5&F/6#, &F\\rF5!#5F5F5F5*&,>*&,*F\\q\"-sIG&)**HF[p\"-WT8Gt6Fbo!+3e<*y\"Ffq\"-' 4G-1Z\"F5Fj[lF5F5F^s\"+%Q3!\\\")*&,0F\\q!)SyASFQ\"*k#[2xFY!*[Q.Z)FE\"* mcqI$F?!)?LgwFbo!+![gf?#F[p!*W,6h*F5F]tF5F5*&,0Fbo\",cq`n\\)F\\q\",WHK AH(FY\",cU7WQ%Ffq\",;?%*p,\"F[p\"-WL$y)Q7FQ\",[+#F[p\"-?R&f@z#Ffq\"-'\\=M$f9FY!+wXm?jF5F euF5F5*&,&F\\q\",CE;\"*[#Ffq\",3AIaE%F5F]vF5F5*&,.FQ!*[\\*RNFbo!*kC?g$ FE!*Q&)3F%F[p!*SY->\"F?!*$QJ%e\"FY!*KM)GLF5FivF5F5*&,0Fbo\",'\\qr[BF 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,0Fbo!*oZaY\"FY!+G0c@;F?!(+%)*pF\\q!*ci(\\:F[p!*oF^'>FQ!+#zJ9D#FE!*W(f IwF5FdzF5F5*&,<*&,0F[p\")K7:8Fbo\")'pH5$FE\")siDmF?\"'?J$*FY\"*W\"Ht:F Q\"*_,c(>F\\q\"(W`A(F5FdzF5F5*&,.F[p\"++Ksc_F5F[yF5F5*&,.FE!('=A`Fbo\"'_VS!&:%)* F5FY\"'7NVFQ!(wj[#F?!(Eew#F5FgyF5F5F5F\\rF5F5F5F\\rF5F5F5F\\rF5F5F5,0F fyF5FQ\"(OZ:\"FE\"(o+<&Fbo!(!o*R\"FY!(kWM#F?\"(Q5m$*&,0Fd`lF5FY\"(%=)H %FQ!(;q6#FE!(e%y%*Fbo\"(!3mDF?!(.>r'*&,0FfyF5FQFe]mFEFf]mFboFg]mFYFh]m F?Fi]m*&,.Fbo\"'!GL#FY\"'W2RFQ!'cC>FE!'y;')F?!'t,hF`]mF5F5F\\rF5F5F5F \\rF5F5F5F\\rF5F5FjyFjy7#-I$seqGF`r6$Fbr/F\\r;F1F^rF%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 22 "proc_qbb_orderNot2 (N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 77 "dev_qbb_orderNot2:=expand(subs(z1=z1/z2,t=t*z2,sum(%[i]*t**(i-1),i =1..N+1))):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 46 "The series q_b lackVertex_whiteVertex_orderNot2" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 2585 "proc_qbw_orderNot2 := proc (n) local i, u; \+ u[0] := 0; u[1] := 0; u[2] := 4*z1; u[3] := 0; u[4] := 20*z1^2; u[5] : = 20*z1^3+20*z1^2; u[6] := 184*z1^3; u[7] := 308*z1^4+308*z1^3; u[8] : = 144*z1^5+2068*z1^4+144*z1^3; u[9] := 4356*z1^5+4356*z1^4; for i from 10 to n do u[i] := expand(factor(-(-2673868800*z1^6*u[i-9]+(8164800*z 1^4+6235380*z1^3+8164800*z1+6235380*z1^2)*u[i-2]+(67415040*z1^2+674150 40*z1^6+5301996480*z1^4+2496525120*z1^3+2496525120*z1^5)*u[i-5]+(-3138 969600*z1^6-3138969600*z1^5)*u[i-8]+(1306368*z1^3+1306368*z1+2774574*z 1^2)*u[i-1]+(65416320*z1^2+452054520*z1^3+452054520*z1^4+65416320*z1^5 )*u[i-4]+(1515525120*z1^4+1515525120*z1^6+647946240*z1^5)*u[i-7]+(-123 922440*z1^4+13970880*z1+13970880*z1^5-296086860*z1^3-123922440*z1^2)*u [i-3]+(2048040960*z1^3+5936438400*z1^4+5936438400*z1^5+2048040960*z1^6 )*u[i-6]+(1005322240*z1^6*u[i-9]+(1036001280*z1^6+1036001280*z1^5)*u[i -8]+(-899696640*z1^6-899696640*z1^4-926376960*z1^5)*u[i-7]+(-310815360 0*z1^5-969666560*z1^3-3108153600*z1^4-969666560*z1^6)*u[i-6]+(-3489512 4*z1^2-16687296*z1-34895124*z1^3-16687296*z1^4)*u[i-2]+(-39290880*z1^6 -1264851072*z1^3-2696814336*z1^4-39290880*z1^2-1264851072*z1^5)*u[i-5] +(-762048*z1-762048*z1^3-1820799*z1^2)*u[i-1]+(-300381264*z1^4-3003812 64*z1^3-55766016*z1^2-55766016*z1^5)*u[i-4]+(-20248704*z1+87630660*z1^ 3+14928624*z1^4+14928624*z1^2-20248704*z1^5)*u[i-3]+((162938880*z1^6+1 62938880*z1^4+222658560*z1^5)*u[i-7]+(529643520*z1^4+152862720*z1^6+15 2862720*z1^3+529643520*z1^5)*u[i-6]-125829120*z1^6*u[i-9]+(7618560*z1^ 6+458924544*z1^4+7618560*z1^2+214914048*z1^3+214914048*z1^5)*u[i-5]+(- 108134400*z1^6-108134400*z1^5)*u[i-8]+(18874368*z1^2+77680512*z1^4+188 74368*z1^5+77680512*z1^3)*u[i-4]+(19979136*z1^4+17532288*z1^3+8709120* z1^5+19979136*z1^2+8709120*z1)*u[i-3]+(20929104*z1^3+8418816*z1+841881 6*z1^4+20929104*z1^2)*u[i-2]+(-54432*z1-54432*z1^3+52974*z1^2)*u[i-1]+ ((-2230272*z1^2-7544448*z1^4-2230272*z1^5-7544448*z1^3)*u[i-4]+(-93388 80*z1^4-9338880*z1^6-14807040*z1^5)*u[i-7]+(-4174848*z1^4-1161216*z1-1 161216*z1^5-4174848*z1^2-5525712*z1^3)*u[i-3]+(-8028160*z1^3-8028160*z 1^6-29521920*z1^5-29521920*z1^4)*u[i-6]+5242880*z1^6*u[i-9]+(-1254528* z1^4-1254528*z1-3330072*z1^2-3330072*z1^3)*u[i-2]+(-491520*z1^2-491520 *z1^6-26281728*z1^4-12343296*z1^5-12343296*z1^3)*u[i-5]+(3440640*z1^6+ 3440640*z1^5)*u[i-8]+(54432*z1^3+54432*z1+81891*z1^2)*u[i-1])*i)*i)*i) /(-734832-734832*z1^3-2042658*z1^2-2042658*z1+(1347192+1347192*z1^3+37 44873*z1^2+3744873*z1+(-734832-734832*z1^3-2042658*z1^2-2042658*z1+(12 2472*z1^3+340443*z1^2+340443*z1+122472)*i)*i)*i))) end do; [seq(u[i], \+ i = 0 .. n)] end proc;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I3proc_qbw_orderNot2G6\"f*6#I\"nGF%6$I\"iGF%I\"uGF%F%F %C.>&8%6#\"\"!F1>&F/6#\"\"\"F1>&F/6#\"\"#,$I#z1GF%\"\"%>&F/6#\"\"$F1>& F/6#F<,$*$F;F9\"#?>&F/6#\"\"&,&*$F;F@FFFEFF>&F/6#\"\"',$FL\"$%=>&F/6# \"\"(,&*$F;F<\"$3$FLFY>&F/6#\"\"),(*$F;FJ\"$W\"FX\"%o?FLFjn>&F/6#\"\"* ,&Fin\"%cVFXFao?(8$\"#5F59$I%trueGI*protectedGFgo>&F/6#Fco-I'expandGFg o6#-I'factorGI(_syslibGF%6#,$*&,6*&F;FP&F/6#,&FcoF5!\"*F5F5!++)oQn#*&, *FX\"(+[;)FL\"(!QNiF;F]qFEF^qF5&F/6#,&FcoF5!\"#F5F5F5*&,,FE\")S]Tn*$F; FPFeqFX\"+!['*>I&FL\"+?^_'\\#FinFhqF5&F/6#,&FcoF5!\"&F5F5F5*&,&Ffq!++' p*QJFinF_rF5&F/6#,&FcoF5!\")F5F5F5*&,(FL\"(ojI\"F;FfrFE\"(uXx#F5&F/6#, &FcoF5!\"\"F5F5F5*&,*FE\")?jTlFL\"*?X0_%FXF_sFinF^sF5&F/6#,&FcoF5!\"%F 5F5F5*&,(FX\"+?^_::FfqFfsFin\"*Si%zkF5&F/6#,&FcoF5!\"(F5F5F5*&,,FX!*SC #R7F;\")!)3(R\"FinF_tFL!*go3'HFEF^tF5&F/6#,&FcoF5!\"$F5F5F5*&,*FL\"+g4 /[?FX\"++%Qk$fFinFhtFfqFgtF5&F/6#,&FcoF5!\"'F5F5F5*&,6Fep\"+SAK05*&,&F fq\"+!G,g.\"FinFbuF5F`rF5F5*&,(Ffq!*Smp**)FXFeuFin!*gpPE*F5FhsF5F5*&,* Fin!++O:3JFL!*glmp*FXFiuFfqFjuF5FitF5F5*&,*FE!)C^*[$F;!)'H(o;FLF]vFXF^ vF5F_qF5F5*&,,Ffq!)!)3HRFL!+s5&[E\"FX!+OV\"op#FEFavFinFbvF5FiqF5F5*&,( F;!'[?wFLFfvFE!(*z?=F5FhrF5F5*&,*FX!*k7Q+$FLFjvFE!);gwbFinF[wF5F`sF5F5 *&,,F;!)/([-#FL\")g1j()FX\")C'G\\\"FEF`wFinF^wF5FatF5F5*&,6*&,(Ffq\"*! ))QH;FXFewFin\"*g&eEAF5FhsF5F5*&,*FX\"*?NkH&Ffq\"*?F'G:FLFjwFinFiwF5Fi tF5F5Fep!*?\"He7*&,,Ffq\"(g&=wFX\"*WX#*e%FEF^xFL\"*[S\"\\@FinF`xF5FiqF 5F5*&,&Ffq!*+W83\"FinFcxF5F`rF5F5*&,*FE\")oV()=FX\")70oxFinFfxFLFgxF5F `sF5F5*&,,FX\")O\"z*>FL\"))GKv\"Fin\"(?\"4()FEFjxF;F\\yF5FatF5F5*&,*FL \")/\"H4#F;\"(;)=%)FXF`yFEF_yF5F_qF5F5*&,(F;!&KW&FLFcyFE\"&uH&F5FhrF5F 5*&,4*&,*FE!(s-B#FX!([Wa(FinFiyFLFjyF5F`sF5F5*&,(FX!(!))Q$*FfqF]zFin!) Sq![\"F5FhsF5F5*&,,FX!([[<%F;!(;7;\"FinFbzFEFazFL!(7d_&F5FatF5F5*&,*FL !(g\"G!)FfqFfzFin!)?>_HFXFgzF5FitF5F5Fep\"(!)GC&*&,*FX!(GXD\"F;F[[lFE! (s+L$FLF\\[lF5F_qF5F5*&,,FE!'?:\\FfqF_[lFX!)GZ 8F5FLF^\\lFE\"(t[u$F;F_\\l*&,,Fj[lF5FLFj[lFEF[\\lF;F[\\l*&,*FL\"'sC7FE \"'V/MF;Fe\\lFd\\lF5F5FcoF5F5F5FcoF5F5F5FcoF5F5F[sF[s7#-I$seqGFgo6$Fio /Fco;F1FeoF%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 22 "proc_qbw_orderNot2(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 75 "dev_qbw_orderNot2:=expand(subs(z1 =z1/z2,t=t*z2,sum(%[i]*t**(i-1),i=1..N))):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 46 "The series q_whiteVertex_whiteVertex_orderNot2" } {TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 57 "dev_qww_or derNot2:=subs(\{z1=z2,z2=z1\},dev_qbb_orderNot2):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 43 "The series q_blackVertex_blackVertex_order2" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 87 "proc_qbb_ order2 := proc (n) local i, u; u[0] := 0; u[1] := 0; u[2] := 0; u[3] : = 0; u[\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4] := 0; u[5] := 0; \+ u[6] := 3*z1^2; u[7] := 14*z1^3; u[8] := 12*z1^4+48*z1^3; u[\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9] := 234*z1^4+30*z1^3; u[10] := 330*z1^5+735*z1^4; u[11] := 143*z1^6+3432*z1^5+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n781*z1^4; u[12] := 6214*z1^6+10932*z1^5+273*z1^4; u[13] := 4914*z1^7+49400*z1^6+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n15288*z1^5; u[14] := 9912*z1^5+1428*z1^8+104020*z1^7+161105*z1^6 ; u[15] := 11322\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^8+7119 30*z1^7+268540*z1^6+2448*z1^5; u[16] := 243168*z1^6+62016*z1^9+165954 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^8+2372640*z1^7; u[17] \+ := 13566*z1^10+2220948*z1^9+10320870*z1^8+4475250*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7+114342*z1^6; u[18] := 5049972*z1^7+172179 9*z1^10+25891110*z1^9+35034615*z1^8+2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1945*z1^6; u[19] := 721050*z1^11+40135011*z1^10+150641120*z1 ^9+72470769*z1^8+339\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2070*z1 ^7; u[20] := 126500*z1^12+39209940*z1^11+399513324*z1^10+519291960*z1^ 9+9\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n5875395*z1^8+1254880*z1^ 7; u[21] := 23580865*z1^12+691816356*z1^11+2213155224*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 96 "\n^10+1154074152*z1^9+81846765*z1^8+197340* z1^7; for i from 22 to n do u[i] :=expand(factor( -(-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n21992838266880*z1^13*u[i-20]+(-282852232815 552*z1^11-17926530045120*z1^12+936913\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n36486272*z1^5+407402659212768*z1^6+4538083415040*z1^4-473051 1669307056*z1^9-1822\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7697383 95008*z1^10-744384073594368*z1^7-4304836927029840*z1^8-54156211200*z1^ 13)\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*u[i-10]+(-1159976161840 0512*z1^11-1977888626964288*z1^12+43900412313600*z1^6-84\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n38865221334816*z1^9-19773960840660192*z1^ 10+956678732190720*z1^7+175416010077120\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n0*z1^8-78052989588480*z1^13)*u[i-13]+(-621974884731 9552*z1^11-6367238062216704*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n1^12+528593788846080*z1^9-1279347167001600*z1^13-191900256076800*z1^ 10)*u[i-16]+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n(-7419047510016 0*z1^12-194690609971200*z1^13)*u[i-19]+(-40810683841920*z1^11+277\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n547212800*z1^3-275793739200*z1^1 2+88897733988000*z1^5+44000755275024*z1^6+124970\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n61156480*z1^4-1223203864661616*z1^9-3436176921547 20*z1^10-890558658412104*z1^7-1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n799237626213224*z1^8)*u[i-9]+(-5046484923531072*z1^11-5344778407 77792*z1^12+4021\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n609512960*z 1^5+258912753177600*z1^6-13475444584591488*z1^9-14796158761276512*z1^ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n10+1514092452757632*z1^7-16 52760687366048*z1^8-18145615057920*z1^13)*u[i-12]+(-1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3599025272654336*z1^11+457657195806720*z1^8 -5348485045358784*z1^10+1423488035773\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n440*z1^9-7115614379060544*z1^12-804534172646400*z1^13)*u[i-1 5]+(-845575736647680\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^12- 15612084633600*z1^11-671897417932800*z1^13)*u[i-18]+(-894930298560*z1^ 11+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n930401942400*z1^3+903598 41600*z1^12+22778234818560*z1^5-78793422136560*z1^6+1189\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n8791857440*z1^4-200528086017120*z1^9-4535 9400246048*z1^10-302961104960688*z1^7-3\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n68966835419136*z1^8)*u[i-8]+(-1402207276828608*z1^1 1-114203441479104*z1^12+37409\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n415828480*z1^5+519492406915584*z1^6+351476858880*z1^4-10667579741054 688*z1^9-660\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1683615052608*z 1^10+622609250997600*z1^7-4935898956811104*z1^8-3204957680640*z1^\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n13)*u[i-11]+(-16303238567615040* z1^11-4787701771122240*z1^12-1050054300938112*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^9-14941952894578176*z1^10+203704472125440*z1^7+1 769914420853760*z1^8-3061834755\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n84000*z1^13)*u[i-14]+(-1217137940167680*z1^13-3270656864624640*z 1^12-12272738563\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n12320*z1^11 +270996337704960*z1^10)*u[i-17]+(247121824320*z1^11+869704040736*z1^3 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-7268666508252*z1^5-5356795 253256*z1^6+1817548770240*z1^4+5966136000*z1^2-219469\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n73259168*z1^9-1148631056640*z1^10-567669222 2268*z1^7-33969935936520*z1^8)*u[i-7]\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+(25415536698*z1^3-584468460*z1+11789188254*z1^5+10324519718 4*z1^6-11048516316*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^4-547 4130336*z1^2+43807998204*z1^7-4363492680*z1^8)*u[i-3]+(-36035301888*z1 ^3+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1449174989952*z1^5+20403 932684688*z1^6-2547927994752*z1^4-6968983392*z1^2-183468\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n62448*z1^9+235372354560*z1^10+15863455560 528*z1^7+233933135184*z1^8)*u[i-6]+(106\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n821828*z1^3-134386776*z1-431145180*z1^5+4051204632* z1^6-5964109380*z1^4+13913839\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n80*z1^2-59923800*z1^7+11022480*z1^8)*u[i-2]+(-333317482992*z1^3+3483 355321980*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^5+673595058769 2*z1^6-598874712516*z1^4-39370990032*z1^2+133552906704*z1^9-61916\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n40000*z1^10+3835757511792*z1^7+9 73860408648*z1^8)*u[i-5]+(236196+30919806*z1^3+3\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4725186*z1-388682388*z1^5-122209560*z1^6-11197440 0*z1^4-91451592*z1^2+22044960*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^7)*u[i-1]+(-55105596216*z1^3-413138880*z1+593332175088*z1^5+94 5878998068*z1^6+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 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"\n305344*z1^7 -212336640*z1^4+973030360960*z1^11+2169569280*z1^13+6814839331624*z1^ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9-324158322688*z1^6+4303147 728216*z1^10-25884426240*z1^5)*u[i-11]+(-682344*z1^7+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7128891*z1^6+19825884*z1^5+259524*z1^4-2698 029*z1^3+4080213*z1^2-948429*z1-39366\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n)*u[i-1])*i)*i)*i)/(236196+12360924*z1^3-2217618*z1-17701578 *z1^5-314928*z1^6-61\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n93584*z 1^4-1285956*z1^2+(-22661694*z1^3+32452893*z1^5+2357586*z1^2-433026+113 549\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n04*z1^4+577368*z1^6+4065 633*z1+(236196+12360924*z1^3-2217618*z1-17701578*z1^5-31\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4928*z1^6-6193584*z1^4-1285956*z1^2+(5248 8*z1^6+2950263*z1^5+1032264*z1^4-206015\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 80 "\n4*z1^3+214326*z1^2+369603*z1-39366)*i)*i)*i))) end \+ do; [seq(u[i], i = 0 .. n)] " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 10 "\ne 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"dev_qbb_order2:=expand(subs(z1=z1/z2,t=t*z2,sum(%[i]*t**(i-1), i=1..N+1))):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 43 "The series q_ whiteVertex_whiteVertex_order2" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 51 "dev_qww_order2:=subs(\{z1=z2,z2=z1\},dev_qbb_or der2):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 43 "The series q_blackV ertex_whiteVertex_order2" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 3300 "proc_qbw_order2 := proc (n) local i, u; u[0] := 0; \+ u[1] := 0; u[2] := 0; u[3] := 0; u[4] := 4*z1^2; u[5] := 0; u[6] := 28 *z1^3; u[7] := 28*z1^4+28*z1^3; u[8] := 276*z1^4; u[9] := 468*z1^5+468 *z1^4; u[10] := 220*z1^6+3224*z1^5+220*z1^4; u[11] := 6864*z1^6+6864*z 1^5; for i from 12 to n do u[i]:=expand(factor( -(-157416161280*z1^7* u[i-11]+(7122330*z1^3+14192172*z1^2+7122330*z1)*u[i-1]+(-2453982228*z1 ^3-2453982228*z1^4-46016520*z1^5-46016520*z1^2)*u[i-4]+(-3286456320*z1 ^3-37422634240*z1^6-41551213376*z1^5-3286456320*z1^7-37422634240*z1^4) *u[i-7]+(-371671236608*z1^6-371671236608*z1^7)*u[i-10]+(163800*z1-1251 62100*z1^2+163800*z1^5-237211956*z1^3-125162100*z1^4)*u[i-3]+(29340989 928*z1^4-3422697600*z1^3+29340989928*z1^5-3422697600*z1^6)*u[i-6]+(-26 0256847872*z1^7-651277676544*z1^6-260256847872*z1^5)*u[i-9]+(14610780* z1+84128544*z1^3+84128544*z1^2+14610780*z1^4)*u[i-2]+(2405225736*z1^3+ 3866189364*z1^4+2405225736*z1^5-202567680*z1^2-202567680*z1^6)*u[i-5]+ (-47626629120*z1^4-327762144384*z1^5-327762144384*z1^6-47626629120*z1^ 7)*u[i-8]+((111396569088*z1^7+111396569088*z1^6)*u[i-10]+(79080501248* z1^7+199174165504*z1^6+79080501248*z1^5)*u[i-9]+(16086540288*z1^4+1008 54403328*z1^6+100854403328*z1^5+16086540288*z1^7)*u[i-8]+47303753728*z 1^7*u[i-11]+(-14710734*z1^2-7350993*z1^3-7350993*z1)*u[i-1]+(107813712 *z1^5+1533856428*z1^3+1533856428*z1^4+107813712*z1^2)*u[i-4]+(12640051 20*z1^3+12090788352*z1^6+10302907392*z1^5+12090788352*z1^4+1264005120* z1^7)*u[i-7]+(107466588*z1^2+107466588*z1^4-209520*z1-209520*z1^5+1969 83603*z1^3)*u[i-3]+(1372870912*z1^3-10514085264*z1^4-10514085264*z1^5+ 1372870912*z1^6)*u[i-6]+(-10443060*z1^4-10443060*z1-63116739*z1^3-6311 6739*z1^2)*u[i-2]+(-486620880*z1^3+122342400*z1^6+122342400*z1^2-35604 1956*z1^4-486620880*z1^5)*u[i-5]+((-161710080*z1^7-1301467136*z1^6-130 1467136*z1^4-749483008*z1^5-161710080*z1^3)*u[i-7]+(-11124736000*z1^6- 11124736000*z1^7)*u[i-10]+(1180053120*z1^4+1180053120*z1^5-189704192*z 1^3-189704192*z1^6)*u[i-6]+(-8043069440*z1^7-8043069440*z1^5-201842769 92*z1^6)*u[i-9]+(-36656512*z1^5-24125440*z1^6-193162848*z1^4-24125440* z1^2-36656512*z1^3)*u[i-5]+(-10278004736*z1^5-1814364160*z1^7-18143641 60*z1^4-10278004736*z1^6)*u[i-8]-4734320640*z1^7*u[i-11]+(4768632*z1^2 +2366820*z1+2366820*z1^3)*u[i-1]+(-322828416*z1^3-36003456*z1^5-360034 56*z1^2-322828416*z1^4)*u[i-4]+(-29405376*z1^2-29405376*z1^4+86400*z1^ 5+86400*z1-53661852*z1^3)*u[i-3]+(2388960*z1+15164010*z1^3+2388960*z1^ 4+15164010*z1^2)*u[i-2]+((22814784*z1^3+3250944*z1^5+3250944*z1^2+2281 4784*z1^4)*u[i-4]+(6881280*z1^3+6881280*z1^7+46977024*z1^4+14405376*z1 ^5+46977024*z1^6)*u[i-7]+(370212864*z1^6+370212864*z1^7)*u[i-10]+(-115 20*z1+4783725*z1^3-11520*z1^5+2574864*z1^2+2574864*z1^4)*u[i-3]+(-3940 6848*z1^4-39406848*z1^5+9097216*z1^3+9097216*z1^6)*u[i-6]+(273874944*z 1^7+273874944*z1^5+677539840*z1^6)*u[i-9]+(-177120*z1-1157571*z1^3-177 120*z1^4-1157571*z1^2)*u[i-2]+(8576000*z1^3+1556480*z1^2+24302928*z1^4 +8576000*z1^5+1556480*z1^6)*u[i-5]+(68354048*z1^4+68354048*z1^7+346763 264*z1^5+346763264*z1^6)*u[i-8]+157810688*z1^7*u[i-11]+(-235953*z1^3-2 35953*z1-480654*z1^2)*u[i-1])*i)*i)*i)/(-7290-7290*z1^3+30618*z1+30618 *z1^2+(-56133*z1+13365*z1^3-56133*z1^2+13365+(-7290-7290*z1^3+30618*z1 +30618*z1^2+(1215*z1^3-5103*z1^2-5103*z1+1215)*i)*i)*i))) end do; [seq (u[i], i = 0 .. n)] end proc;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I0proc_qbw_order2G6\"f*6#I\"nGF%6$I\"iGF%I\"uGF%F%F%C0 >&8%6#\"\"!F1>&F/6#\"\"\"F1>&F/6#\"\"#F1>&F/6#\"\"$F1>&F/6#\"\"%,$*$I# z1GF%F9FA>&F/6#\"\"&F1>&F/6#\"\"',$*$FDF=\"#G>&F/6#\"\"(,&*$FDFAFOFNFO >&F/6#\"\"),$FU\"$w#>&F/6#\"\"*,&*$FDFH\"$o%FUF\\o>&F/6#\"#5,(*$FDFL\" $?#F[o\"%CKFUFco>&F/6#\"#6,&Fbo\"%koF[oFjo?(8$\"#7F59$I%trueGI*protect edGF`p>&F/6#F\\p-I'expandGF`p6#-I'factorGI(_syslibGF%6#,$*&,:*&FDFS&F/ 6#,&F\\pF5!#6F5F5!-!GhhTd\"*&,(FN\"(IB7(FC\")s@>9FDFfqF5&F/6#,&F\\pF5! \"\"F5F5F5*&,*FN!+GA)RX#FUF^rF[o!)?l,YFCF_rF5&F/6#,&F\\pF5!\"%F5F5F5*& ,,FN!+?jX'G$Fbo!,SUjAu$F[o!,wL@^:%*$FDFSFfrFUFgrF5&F/6#,&F\\pF5!\"(F5F 5F5*&,&Fbo!-3mBr;PFirF`sF5&F/6#,&F\\pF5!#5F5F5F5*&,,FD\"'+Q;FC!*+@;D\" F[oFgsFN!*c>@P#FUFhsF5&F/6#,&F\\pF5!\"$F5F5F5*&,*FU\",G**)4MHFN!++wpAM F[oF`tFboFatF5&F/6#,&F\\pF5!\"'F5F5F5*&,(Fir!-sy%oDg#Fbo!-Wlnx7lF[oFht F5&F/6#,&F\\pF5!\"*F5F5F5*&,*FD\")!y5Y\"FN\")W>)FCFauFUF`uF5&F/6#,&F \\pF5!\"#F5F5F5*&,,FN\"+OdA0CFU\"+k$*=mQF[oFhuFC!*!onD?FboFjuF5&F/6#,& F\\pF5!\"&F5F5F5*&,*FU!,?\"HmiZF[o!-%QW@wF$FboFbvFirFavF5&F/6#,&F\\pF5 !\")F5F5F5*&,:*&,&Fir\"-)3plR6\"FboF[wF5FasF5F5*&,(Fir\",[7]!3zFbo\"-/ b;u\"*>F[oF^wF5FjtF5F5*&,*FU\",)GSl3;Fbo\"-GLSa35F[oFcwFirFbwF5FcvF5F5 F^q\",GPv.t%*&,(FC!)M2r9FN!($*4N(FDFhwF5FhqF5F5*&,*F[o\"*7P\"y5FN\"+Gk &Q`\"FUF\\xFCF[xF5F`rF5F5*&,,FN\"+?^+k7Fbo\",_$)y!47F[o\",#R2HI5FUF`xF irF_xF5FjrF5F5*&,,FC\"*)emu5FUFdxFD!'?&4#F[oFexFN\"*.O)p>F5FjsF5F5*&,* FN\"+74(GP\"FU!,k_390\"F[oFjxFboFixF5FbtF5F5*&,*FU!)gIW5FDF]yFN!)Rn6jF CF^yF5FbuF5F5*&,,FN!*!)3i'[Fbo\"*+CMA\"FCFbyFU!*c>/c$F[oFayF5F[vF5F5*& ,:*&,,Fir!*!35<;Fbo!+OrY,8FUFiyF[o!*3I[\\(FNFhyF5FjrF5F5*&,&Fbo!,+gtC6 \"FirF]zF5FasF5F5*&,*FU\"+?J0!=\"F[oF`zFN!*#>/(*=FboFazF5FbtF5F5*&,(Fi r!+S%pI/)F[oFdzFbo!,#*pF%=?F5FjtF5F5*&,,F[o!)7llOFbo!)Sa7CFU!*[G;$>FCF izFNFhzF5F[vF5F5*&,*F[o!,OZ+y-\"Fir!+gTO9=FUF^[lFboF][lF5FcvF5F5F^q!+S 1KMZ*&,(FC\"(K'oZFD\"(?oO#FNFc[lF5FhqF5F5*&,*FN!*;%GGKF[o!)cM+OFCFg[lF UFf[lF5F`rF5F5*&,,FC!)w`SHFUFj[lF[o\"&+k)FDF[\\lFN!)_=m`F5FjsF5F5*&,*F D\"(g*)Q#FN\")5S;:FUF_\\lFCF`\\lF5FbuF5F5*&,8*&,*FN\")%y9G#F[o\"(W4D$F CFf\\lFUFe\\lF5F`rF5F5*&,,FN\"(!G\")oFirFi\\lFU\")Cq(p%F[o\")w`S9FboFj \\lF5FjrF5F5*&,&Fbo\"*kG@q$FirF^]lF5FasF5F5*&,,FD!&?:\"FN\"(DPy%F[oFa] lFC\"(k[d#FUFc]lF5FjsF5F5*&,*FU!)[oSRF[oFf]lFN\"(;s4*FboFg]lF5FbtF5F5* &,(Fir\"*W\\(QFF[oFj]lFbo\"*S)RvnF5FjtF5F5*&,*FD!'?r " 0 "" {MPLTEXT 1 0 19 "proc_qbw_order2(N):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 74 "dev_qbw_order2:=expand(sub s(z1=z1/z2,t=t*z2,sum(%[i]*t**(i-1),i=1..N+1))):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 35 "The series q_rootedFace_blackVertex" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 81 "proc_q_rooted_bl ackVertex := proc (n) local i, u; u[0] := 0; u[1] := 0; u[2] := \\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0; u[3] := 0; u[4] := z1^2*z2^2; u[5] := z1^3*z2^2; u[6] := 6*z1^3*z2^3; u[7] :=\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n 12*z1^4*z2^3+4*z1^3*z2^4; u[8] := 6*z1^5*z2^3+51 *z1^4*z2^4; u[9] := 129*z1^5*z2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^4+60*z1^4*z2^5; u[10] := 121*z1^6*z2^4+512*z1^5*z2^5+22*z1^4*z2 ^6; u[11] := 39*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^7*z2^4+1 448*z2^5*z1^6+794*z1^5*z2^6; u[12] := 1872*z1^7*z2^5+5635*z1^6*z2^6+5 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n46*z1^5*z2^7; u[13] := 1148 *z1^8*z2^5+17035*z1^7*z2^6+10360*z1^6*z2^7+140*z1^5*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 96 "\n2^8; for i from 14 to n do u[i] := expand(f actor(-(-6104678400*z1^10*z2^7*u[i-13]+(-45719856*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^2*z2^5+66721536*z1^4*z2^3+68081472*z1^5*z2^ 2+393120*z1*z2^6-20078064*z1^3*z2^4\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+19279620*z1^6*z2)*u[i-3]+(8754614400*z1^4*z2^6-3030754560*z1^ 5*z2^5-13406038368\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^6*z2^ 4-767784240*z1^8*z2^2-5918193600*z1^7*z2^3+1437004800*z2^7*z1^3)*u[i-6 ]\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+(46197895680*z1^6*z2^7-27 668883552*z1^7*z2^6-45643877280*z1^9*z2^4-134303354880\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^8*z2^5-1020418560*z1^10*z2^3)*u[i-9]+(- 22078586880*z1^10*z2^6-18882109440*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^9*z2^7)*u[i-12]+(6422490*z1^5*z2+3257496*z1^2*z2^4-2599776* z1*z2^5+4503924*z1^3\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z2^3+5 050188*z1^4*z2^2)*u[i-2]+(-946514016*z1^4*z2^5-1681801704*z1^5*z2^4+11 572\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n10208*z2^6*z1^3-77818615 2*z1^6*z2^3-461982240*z1^7*z2^2+100154880*z2^7*z1^2)*u[i\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-5]+(30692874240*z1^5*z2^7+28485921600*z1 ^6*z2^6-86708917008*z1^7*z2^5-662143714\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n08*z1^8*z2^4-5737127040*z1^9*z2^3)*u[i-8]+(-1081278 7200*z1^8*z2^7-76177059840*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n^9*z2^6-26704995840*z1^10*z2^5)*u[i-11]+(133650*z1^4*z2-9720*z1*z2^4 +144828*z1^2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z2^3+231336*z1 ^3*z2^2)*u[i-1]+(-168547032*z1^4*z2^4+84813120*z1^2*z2^6+12990780\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^7*z2-420512400*z1^3*z2^5+335 963088*z1^5*z2^3+38390220*z1^6*z2^2)*u[i-4]+(969\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0831360*z2^7*z1^4-330674400*z1^9*z2^2-44191438464 *z1^7*z2^4-25179250032*z2^5*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6-9583115280*z1^8*z2^3+27430711392*z1^5*z2^6)*u[i-7]+(-117515059 20*z1^10*z2^4+25\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n823646720*z 1^7*z2^7-97017050880*z1^9*z2^5-90504390912*z1^8*z2^6)*u[i-10]+((34535 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n11680*z1^10*z2^4+2719461632 0*z1^8*z2^6+29511332352*z1^9*z2^5-9331015680*z1^7*z2^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7)*u[i-10]+1527939072*z1^10*z2^7*u[i-13]+(- 16528860*z1^6*z2-502848*z1*z2^6-49912\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n848*z1^5*z2^2+10345104*z1^3*z2^4-42471486*z1^4*z2^3+31838400 *z1^2*z2^5)*u[i-3]+(\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-395321 9584*z1^4*z2^6+2944937996*z1^7*z2^3+1560944208*z1^5*z2^5+6180238668*z1 ^6*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz2^4-710344704*z2^7*z1^3 +403178544*z1^8*z2^2)*u[i-6]+(14865585984*z1^9*z2^4-1644\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0775680*z1^6*z2^7+334347264*z1^10*z2^3+44 706643968*z1^8*z2^5+8288244032*z1^7*z2^\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n6)*u[i-9]+(4786679808*z1^9*z2^7+5875200000*z1^10*z2 ^6)*u[i-12]+(-4457430*z1^3*z2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n^3-4911354*z1^4*z2^2-3903228*z1^2*z2^4-6852357*z1^5*z2+2450736*z1*z2 ^5)*u[i-2]+(\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n549022848*z1^4* z2^5-619885632*z2^6*z1^3+530033736*z1^6*z2^3+278387460*z1^7*z2^2+\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n929331228*z1^5*z2^4-58752000*z2^ 7*z1^2)*u[i-5]+(2190337920*z1^9*z2^3+31870072752\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^7*z2^5-10947938304*z1^6*z2^6+23945254320*z1^8 *z2^4-11672875008*z1^5*z2^7)*u[\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\ni-8]+(21578840064*z1^9*z2^6+2143558656*z1^8*z2^7+7466425344*z1^1 0*z2^5)*u[i-11]+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n(-201690*z1 ^2*z2^3+13122*z1^5-233523*z1^4*z2-450036*z1^3*z2^2+30132*z1*z2^4)*u[i \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-1]+(-9893988*z1^7*z2+24926 3568*z1^3*z2^5+115288920*z1^4*z2^4-14169546*z1^6*z2^2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-167067306*z1^5*z2^3-56166912*z1^2*z2^6)*u[ i-4]+(-11134204096*z1^5*z2^6-40995256\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n32*z2^7*z1^4+17885103744*z1^7*z2^4+10440133056*z2^5*z1^6+414 2069600*z1^8*z2^3+15\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6064320 *z1^9*z2^2)*u[i-7]+((-599292096*z1^8*z2^3+1499893248*z1^5*z2^6-2426357 568\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^7*z2^4+580116480*z2^ 7*z1^4-1446093504*z2^5*z1^6-24261120*z1^9*z2^2)*u[i-7]+(\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-2999531520*z1^9*z2^5-338411520*z1^10*z2^ 4-2739443712*z1^8*z2^6+1099530240*z1^7*\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\nz2^7)*u[i-10]-127401984*z1^10*z2^7*u[i-13]+(-663033 6*z1^2*z2^5+207360*z1*z2^6+46\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n45674*z1^6*z2+7424946*z1^4*z2^3-748440*z1^3*z2^4+11485800*z1^5*z2^2) *u[i-3]+(593\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n983488*z1^4*z2^ 6+117080064*z2^7*z1^3-956011896*z1^6*z2^4-266160960*z1^5*z2^5-701\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n63136*z1^8*z2^2-491551512*z1^7*z 2^3)*u[i-6]+(1948262400*z1^6*z2^7-36495360*z1^10\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z2^3-4974326784*z1^8*z2^5-836860416*z1^7*z2^6-16 19735040*z1^9*z2^4)*u[i-9]+(-52\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1994240*z1^10*z2^6-404029440*z1^9*z2^7)*u[i-12]+(1308798*z1^3*z2 ^3+1535760*z1^2*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz2^4+143807 4*z1^4*z2^2+2303154*z1^5*z2-668736*z1*z2^5)*u[i-2]+(111002112*z2^6*z1 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^3-55851804*z1^7*z2^2-11909 2968*z1^6*z2^3-173899404*z1^5*z2^4-102835584*z1^4*z2^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n5+11427840*z2^7*z1^2)*u[i-5]+(-279309312*z1 ^9*z2^3+1384611840*z1^6*z2^6-29018864\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n64*z1^8*z2^4-3913376256*z1^7*z2^5+1483800576*z1^5*z2^7)*u[i- 8]+(-696508416*z1^10\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z2^5-2 044624896*z1^9*z2^6-116023296*z1^8*z2^7)*u[i-11]+(116397*z1^4*z2-19683 *z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^5+270216*z1^3*z2^2-2818 8*z1*z2^4+49086*z1^2*z2^3)*u[i-1]+(22185000*z1^5*z2^3+24\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n34860*z1^7*z2+12460032*z1^2*z2^6-46657152 *z1^3*z2^5-25985772*z1^4*z2^4+355032*z1\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n^6*z2^2)*u[i-4]+((-930816*z1^2*z2^6-350262*z1^5*z2^ 3+201438*z1^6*z2^2-193428*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n7*z2+2657088*z1^3*z2^5+1916460*z1^4*z2^4)*u[i-4]+(29033344*z1^8*z2^3 -67011584*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^5*z2^6-2747596 8*z2^7*z1^4+66965760*z2^5*z1^6+1244160*z1^9*z2^2+110477856*z1^7*z\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2^4)*u[i-7]+(11059200*z1^10*z2^4 +101864448*z1^9*z2^5+92578816*z1^8*z2^6-42516480\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^7*z2^7)*u[i-10]+3538944*z1^10*z2^7*u[i-13]+(- 27648*z1*z2^6+366336*z1^2*z2^5-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n802764*z1^5*z2^2-426222*z1^6*z2-157248*z1^3*z2^4-227448*z1^4*z2^ 3)*u[i-3]+(15031\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n488*z1^5*z2 ^5-29717504*z1^4*z2^6+4047552*z1^8*z2^2-6438912*z2^7*z1^3+49730316*z1 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^6*z2^4+27544348*z1^7*z2^3) *u[i-6]+(28687360*z1^7*z2^6+1327104*z1^10*z2^3+590653\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n44*z1^9*z2^4+185037312*z1^8*z2^5-76922880*z 1^6*z2^7)*u[i-9]+(15482880*z1^10*z2^6\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+11354112*z1^9*z2^7)*u[i-12]+(-103032*z1^3*z2^3-199260*z1^2* z2^4-241299*z1^5*z2-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n122472* z1^4*z2^2+46656*z1*z2^5)*u[i-2]+(-6663168*z2^6*z1^3+3732264*z1^7*z2^2+ 886\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1256*z1^6*z2^3-737280*z2 ^7*z1^2+6170112*z1^4*z2^5+11051712*z1^5*z2^4)*u[i-5]+(-5\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7621504*z1^6*z2^6+117935808*z1^8*z2^4+118 97856*z1^9*z2^3-63062016*z1^5*z2^7+1606\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n15872*z1^7*z2^5)*u[i-8]+(64806912*z1^9*z2^6+2167603 2*z1^10*z2^5+933888*z1^8*z2^7\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n)*u[i-11]+(6561*z1^5-51516*z1^3*z2^2-16524*z1^4*z2+7776*z1*z2^4+7776 *z1^2*z2^3)*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nu[i-1])*i)*i)*i )/((-8748*z1^2*z2^2+26244*z1*z2^3+5832*z2^4-16038*z1^3*z2+13122*z\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^4+(24057*z1^3*z2-39366*z1*z2^3 +13122*z1^2*z2^2-8748*z2^4-19683*z1^4+(2916*z2^4\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 84 "\n-4374*z1^2*z2^2+13122*z1*z2^3+6561*z1^4-8019*z1^3 *z2)*i)*i)*i))) end do; [seq(u[i]\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 25 "\n, i = 0 .. n)] end proc;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I:proc_q_rooted_blackVertexG6\"f*6#I\"nGF%6$I\"iGF%I \"uGF%F%F%C2>&8%6#\"\"!F1>&F/6#\"\"\"F1>&F/6#\"\"#F1>&F/6#\"\"$F1>&F/6 #\"\"%*&I#z1GF%F9I#z2GF%F9>&F/6#\"\"&*&FCF=FDF9>&F/6#\"\"',$*&FCF=FDF= FM>&F/6#\"\"(,&*&FCF=FDFAFA*&FCFAFDF=\"#7>&F/6#\"\"),&*&FCFAFDFA\"#^*& FCFHFDF=FM>&F/6#\"\"*,&*&FCFAFDFH\"#g*&FCFHFDFA\"$H\">&F/6#\"#5,(*&FCF AFDFM\"#A*&FCFHFDFH\"$7&*&FDFAFCFM\"$@\">&F/6#\"#6,(*&FCFHFDFM\"$%z*&F DFHFCFM\"%[9*&FDFAFCFS\"#R>&F/6#FW,(*&FCFHFDFS\"$Y&*&FCFMFDFM\"%Nc*&FC FSFDFH\"%s=>&F/6#\"#8,**&FCFHFDFen\"$S\"*&FCFMFDFS\"&g.\"*&FCFSFDFM\"& Nq\"*&FCFenFDFH\"%[6?(8$\"#9F59$I%trueGI*protectedGFer>&F/6#Far-I'expa ndGFer6#-I'factorGI(_syslibGF%6#,$*(,>*(FCFfoFDFS&F/6#,&FarF5!#8F5F5!+ +%yY5'*&,.*&FCFMFDF5\")?'z#>FU!)k!y+#*&FCFHFDF9\")s93oFV\")O:sm*&FDFHF CF9!)c)>d%*&FDFMFCF5\"'?JRF5&F/6#,&FarF5!\"$F5F5F5*&,.*&FDFSFCF=\"++[+ P9*&FDF9FCFen!*SUyn(F\\p!,o$QgS8Fho\"++Wha()Fjo!+gXvII*&FDF=FCFS!++O>= fF5&F/6#,&FarF5!\"'F5F5F5*&,,Fjq\",!o&*y>YF^r!-!)[N.V8F\\r!,_N))ow#*&F CF]oFDFA!,!GxQkX*&FCFfoFDF=!+g&=/-\"F5&F/6#,&FarF5!\"*F5F5F5*&,&*&FCF] oFDFS!,S%4@))=*&FCFfoFDFM!,!)oey?#F5&F/6#,&FarF5!#7F5F5F5*&,,*&FCF5FDF H!(w(*f#*&FCFAFDF9\"()=]]*&FCFHFDF5\"(!\\AkFO\"(CR]%*&FCF9FDFA\"('\\dK F5&F/6#,&FarF5!\"#F5F5F5*&,.*&FCFSFDF9!*SA)>YF_o!*;S^Y**&FDFMFCF=\"+3- @d6*&FCFMFDF=!*_h=y(*&FDFSFCF9\"*!)[:+\"Fao!+/WFcp\",#R62VF*&FDF=FCFen!+!G:Je**&FDF9FCF]o!* +WnI$F5&F/6#,&FarF5!\"(F5F5F5*&,**&FCFenFDFM!,74R/0**&FCFfoFDFA!,?f]^< \"*&FCFSFDFS\",?nkBe#*&FCF]oFDFH!,!)30*& ,*F[^l\",_BL6&HFi]l!+!o:5L*Fg]l\"+!o6NX$Fe]l\",?jh%>FF5F]^lF5F5Fcs\"+s !Rz_\"*&,.F^t!)[G\"*\\F[t!)g)Gl\"Fat\")+%Q=$FV!)'[rC%Fct!'[G]FU\")/^M5 F5FetF5F5*&,.F]u\"*W&yJSFjo\"+3U%4c\"F\\p\"+o'Q-='Fbu\"+'*z$\\%HFho!+% e>K&RF[u!*/ZM5(F5FduF5F5*&,,F^r\",oRk1Z%F\\r\"+KSC)G)F]v\",%)fel[\"F_v \"*ksMM$Fjq!,!ov2W;F5FavF5F5*&,&Fiv\"+++?veFgv\"+3)zmy%F5F[wF5F5*&,,FO !(IuX%Fcw!(a8\"\\Fhw!(GK!RFew!(dB&oFaw\"(O2X#F5FjwF5F5*&,.Fgx!)+?veFex \"*OP.I&Fcx!*Kc))>'F`x\"*guQy#F_o\"*[G-\\&Fao\"*G7LH*F5FjxF5F5*&,,Fay \",?VDXR#Fdy\"+?zL!>#Faq\",_F2q=$F_q!,/$Qz%4\"F]q!,3](Gn6F5FgyF5F5*&,( F_z\"+W`UmuFaz\"+c'eN9#F]z\",k+%)y:#F5FczF5F5*&,,*$FCFH\"&AJ\"F\\[l\"& K,$FI!'O+XF^[l!'BNBFjz!'!p,#F5F`[lF5F5*&,.Ff[l!)Y&pT\"F[\\l!())R*)*Fgn \"*?*)G:\"Fi[l!)7p;cF]\\l\"*oNE\\#Fin!*1t1n\"F5F`\\lF5F5*&,.Fg\\l!+Kc_ *4%Fcp!,'4/U86Fep\",cI8S/\"F[]l\"++'p?9%Fgp\",WP5&)y\"F]]l\"*?V1c\"F5F _]lF5F5*&,>*&,.Fep!+/N4Y9Fg\\l\"*![;,eFgp!+ovNECF]]l!)?6ECF[]l!*'4#H*f Fcp\"+[K*)*\\\"F5F_]lF5F5*&,*Fi]l\"+S-`*4\"F[^l!+?:`**HFe]l!+7PWRFFg]l !*?:TQ$F5F]^lF5F5Fcs!*%)>SF\"*&,.F[t\"(uck%FV\"(Y\\U(Fat!(O.j'FU!'S%[( F^t\")+e[6Fct\"'gt?F5FetF5F5*&,.Fho\"*)[$)RfF[u\"*k+3<\"Fjo!*g4;m#F\\p !*'*=,c*F]u!)OJ;qFbu!*7:b\"\\F5FduF5F5*&,,F^r!+%yEV(\\F_v!)g`\\OF\\r!* ;/'o$)F]v!+S]t>;Fjq\"++CE[>F5FavF5F5*&,&Fgv!*S%HSSFiv!*SU*>_F5F[wF5F5* &,,Fhw\"(gd`\"FO\"()z38Fcw\"(u!Q9Faw!'O(o'Few\"(aJI#F5FjwF5F5*&,.F`x!) /=&e&Fcx\"*7@+6\"Fgx\")SyU6Fao!*/%**Q\"F5FjxF5F5*& ,,Fay!+kk)=!HFdy!*7$4$z#F_q\"+S=h%Q\"F]q\"+w0!Q[\"Faq!+ciP8RF5FgyF5F5* &,(F]z!+'*[iW?F_z!*;%3lpFaz!*'HBg6F5FczF5F5*&,,F\\[l!&)=GFI\"';-FFjz\" &'3\\F^[l\"'(R;\"Fbbl!&$o>F5F`[lF5F5*&,.Fin\")+]=AFf[l\"'K]NFgn!)sd)f# Fi[l\")K+Y7F[\\l\"(g[V#F]\\l!)_rlYF5F`\\lF5F5*&,<*&,.Fi[l!';3$*F[\\l!' GM>F]\\l\"()3dEFin!'i-NFgn\"(gk\">Ff[l\"'Q9?F5F`\\lF5F5*&,.Fep\")gd'p' F]]l\"(gTC\"Fg\\l!)ofZFF[]l\")WL.HFgp\"*cyZ5\"Fcp!)%e6q'F5F_]lF5F5*&,* Fi]l!)![;D%F[^l\"*[W'=5Fe]l\");)yD*Fg]l\")+#f5\"F5F]^lF5F5Fcs\"(W*QN*& ,.F^t!'kF!)Fat\"'OjOFU!'[s:FV!'[uAFct!&[w#F[t!'AiUF5FetF5F5*&,.F[u!(7* QkFjo\"))[J]\"F]u\"(_v/%F\\p\");.t\\Fho!)/vrHFbu\")[VaFF5FduF5F5*&,,F^ r\"*7t.&=F\\r\")gtoGFjq!)!)G#p(F]v\")W`1fF_v\"(/rK\"F5FavF5F5*&,&Fiv\" )!)G[:Fgv\")7TN6F5F[wF5F5*&,,Fhw!'g#*>Fcw!'sC7Faw\"&cm%FO!'KI5Few!'*HT #F5FjwF5F5*&,.F`x\"(kAt$Fcx!(oJm'Fgx!'!GP(Fao\")7<06F_o\"(7,<'Fex\"(c7 '))F5FjxF5F5*&,,Fdy\")cy*=\"F]q!);?1jF_q!)/:idFaq\"*sehg\"Fay\"*3e$z6F 5FgyF5F5*&,(F_z\")Kgn@F]z\")7p!['Faz\"'))Q$*F5FczF5F5*&,,FI!&;:&Fjz\"% wxF\\[lF`^mFbbl\"%hlF^[l!&Cl\"F5F`[lF5F5F5FarF5F5F5FarF5F5F5FarF5F5F5, .*&FCF=FDF5!&Qg\"*&FCF5FDF=\"&Wi#FB!%[()*$FDFA\"%Ke*$FCFAFcbl*&,.Fi^mF h^mFf^m!&m$RFd^m\"&dS#FBFcblF[_mFehl*&,,FB!%uVFd^m!%>!)Ff^mFcblF[_mFa^ mFi^m\"%;HF5FarF5F5F5FarF5F5Fc[lFarFc[lFc[l7#-I$seqGFer6$Fgr/Far;F1Fcr F%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 29 " proc_q_rooted_blackVertex(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 287 "> " 0 "" {MPLTEXT 1 -1 54 "dev_q_rooted_blackVertex:=sum(%[i]*t**(i-1 ),i=1..N+1):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 38 "The series q_ nonRootedFace_blackVertex" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 81 "proc_q_nonRooted_blackVertex := proc (n) local i, u; u[0] := 0; u[1] := 0; u[2] \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n:= 0; u[3] := 0; u[4] := 8*z1^2; u[5] := 10*z1^3; u[6] := 72*z1^3; u [7] := 168*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^4+56*z1^3; u[ 8] := 96*z1^5+816*z1^4; u[9] := 2322*z1^5+1080*z1^4; u[10] := 242\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^6+10240*z1^5+440*z1^4; u[11 ] := 858*z1^7+31856*z1^6+17468*z1^5; u[12] := 44\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n928*z1^7+135240*z1^6+13104*z1^5; u[13] := 29848*z 1^8+442910*z1^7+269360*z1^6+364\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^5; u[14] := 7616*z1^9+753200*z1^8+1844472*z1^7+279440*z1^6; u[15] := 701760\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 96 "\n*z1^9+62478 00*z1^8+4084200*z1^7+146880*z1^6; for i from 16 to n do u[i] :=expand( factor( -(190\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n890639360*z1^1 3*u[i-16]+(-6128179200*z1^3+77842424440*z1^6+51679261120*z1^5+1107\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n37071592*z1^8+68974405072*z1^9- 15116595968*z1^7-68050928000*z1^4-7207143720*z1^1\\" }{MPLTEXT 1 0 0 " " }{MPLTEXT 1 0 82 "\n0)*u[i-6]+(-1626466897920*z1^6+7607848992352*z1^ 8+11028371022752*z1^9-4352118147\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n200*z1^7+677295126240*z1^11+3385796027360*z1^10+18893327040*z1^1 2)*u[i-9]+(40763\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n59027200*z1 ^12+5395108538880*z1^10-9296862013440*z1^9+17662700765184*z1^11+53910 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n420480*z1^13)*u[i-12]+(9998 49590784*z1^13+477580099584*z1^12)*u[i-15]+(-68247479\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n04*z1^3-84004565808*z1^6-13242167280*z1^5+1 6860107856*z1^8-4917551688*z1^9-20624\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n428200*z1^7+14644916640*z1^4-200309760*z1^2+160919136*z1^10) *u[i-5]+(-1747377920\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n640*z1^ 6-362302617600*z1^5+4869620525976*z1^8+2523324988752*z1^9+212767495968 0*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^7+67381082784*z1^11+79 7528115384*z1^10-2390472000*z1^12)*u[i-8]+(-178328304864\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^9+10559801676480*z1^11+677267138880* z1^12+55327380480*z1^13-8660597483520*z\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n1^8+22075605378240*z1^10)*u[i-11]+(4608363479040*z1 ^12-926440243200*z1^11+142964\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n0478720*z1^13)*u[i-14]+(3710194752*z1^3-12501844836*z1^6-18222906840 *z1^5-361317\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0690*z1^8+29454 3540*z1^9-3523845300*z1^7-125486088*z1^4-277226880*z1^2)*u[i-4]+(\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n358693234464*z1^6-423847257600*z 1^5+828071132832*z1^8+483217324992*z1^9+11213040\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n31680*z1^7-58540922880*z1^4-6731134560*z1^11+9999 6227328*z1^10)*u[i-7]+(-5714470\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n070400*z1^8+16529675099040*z1^9-4779188914176*z1^7+2262766528224 *z1^11+142976966\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n89728*z1^10 +928392192*z1^13+298721805120*z1^12)*u[i-10]+(-5311805813760*z1^10+62 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4544680960*z1^13+7747918525 440*z1^12+8039518172160*z1^11)*u[i-13]+(1310400*z1+56\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3393376*z1^3-2222132472*z1^6-329600736*z1^5 +100440972*z1^8-2019043800*z1^7-19384\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3008*z1^4+385328160*z1^2)*u[i-3]+(15616800*z1+198245232*z1^3 -489088530*z1^6-2088\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n23184*z 1^5+229635*z1^8-45528966*z1^7+419134824*z1^4+29279880*z1^2)*u[i-2]+(45 295\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n20*z1+28640952*z1^3-1880 2368*z1^6-34666380*z1^5+459270*z1^7+18310536*z1^4-820368\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^2)*u[i-1]+(-38206439424*z1^13*u[i-16] +(-69870849708*z1^8-26142637664*z1^6+29\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n896491008*z1^4-10506303148*z1^7+2957279232*z1^3-351 82072640*z1^9-11127735744*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n5+3105510168*z1^10)*u[i-6]+(-1116279275600*z1^10-220826891936*z1^11- 238892071577\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6*z1^8+52191093 9648*z1^6+1279297666560*z1^7-3386864656112*z1^9-6113837952*z1^12)\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*u[i-9]+(-91378286592*z1^12-2109 25780992*z1^13)*u[i-15]+(3494323712*z1^3+3638883\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7280*z1^6+9574210744*z1^5-12374495384*z1^8+197289 5340*z1^9-239390112*z1^7-420041\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9328*z1^4+117504000*z1^2-78096312*z1^10)*u[i-5]+(-1648871064424* z1^8+13211362099\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2*z1^5-2956 52519496*z1^10-24637212448*z1^11-918486071664*z1^9+574560782336*z1^6- \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n711654817024*z1^7+972155520 *z1^12)*u[i-8]+(247968473088*z1^11-1031140147200*z1^1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2-304034291712*z1^13)*u[i-14]+(1569448602*z 1^8+4741793058*z1^6+366182190*z1^7-14\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4127836*z1^9-1599627264*z1^3+180741120*z1^2+9752428428*z1^5+ 724385952*z1^4)*u[i-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4]+(-35 1252167464*z1^8-416563150904*z1^7-212617180312*z1^9+24579227648*z1^4-1 148\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n54019584*z1^6-4225857106 4*z1^10+160059028224*z1^5+2873954016*z1^11)*u[i-7]+(1514\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n807214080*z1^8-693139523872*z1^11-4788353 368928*z1^9+1392584777728*z1^7-87059899\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n008*z1^12-4004400453760*z1^10-274212864*z1^13)*u[i- 10]+(1325743757312*z1^10-1335\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n13574400*z1^13-1773766281216*z1^12-1923989469184*z1^11)*u[i-13]+(819 08136*z1^5-4\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2855156*z1^8+11 35287144*z1^7+1179666648*z1^6-195366528*z1^2+150535800*z1^4-33363\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7128*z1^3-1676160*z1)*u[i-3]+(-9 58620668928*z1^12-16113973248*z1^13-142880839577\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6*z1^10+2393969369088*z1^9-4361664856064*z1^11)*u [i-12]+(142907868*z1^5-352107*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^8+41643639*z1^7+369458091*z1^6-29452032*z1^2-327307608*z1^4-14 0246964*z1^3-115\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n81920*z1)*u [i-2]+(-2709634423936*z1^11+2350413938688*z1^8-5896509476224*z1^10-19 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n7636718720*z1^12+3515625934 08*z1^9-14543456256*z1^13)*u[i-11]+(36896472*z1^5+198\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n60390*z1^6-527067*z1^7-30139776*z1^3-195326 64*z1^4+1303776*z1^2-4552200*z1)*u[i-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1]+(2548039680*z1^13*u[i-16]+(-475103232*z1^3+2356779272*z1^ 6+5943930248*z1^9-43\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9684710 4*z1^4+13450159952*z1^8+52387712*z1^5+3512919608*z1^7-434750664*z1^10) *u[\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\ni-6]+(14854127616*z1^13+ 5786566656*z1^12)*u[i-15]+(-5442833068*z1^6-603996160*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^3-2138128712*z1^5-22855680*z1^2-22297257 6*z1^9+242269824*z1^4+2795335438*z1^8+1\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n728476334*z1^7+11335464*z1^10)*u[i-5]+(-21442494464 *z1^11+21514715136*z1^13+7723\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n7452800*z1^12)*u[i-14]+(21665556*z1^9-39290880*z1^2-495039492*z1^6-1 83654774*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n8-1815838104*z1 ^5+211464576*z1^3-248499240*z1^4+289126956*z1^7)*u[i-4]+(59295293\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n04*z1^10-20280264704*z1^5+311331 77508*z1^9-3443916800*z1^4+48986278212*z1^8-4070\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n27328*z1^11+11639973248*z1^6+51253195928*z1^7)*u[ i-7]+(376570959360*z1^10+704478\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n91328*z1^11-132915523584*z1^8+8478704640*z1^12+463612509824*z1^9 -135872118784*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^7+26984448 *z1^13)*u[i-10]+(9524920320*z1^13-110050131968*z1^10+154680786944*z1^ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n11+135435681792*z1^12)*u[i- 13]+(31003776*z1^2-515592*z1^5-179829828*z1^6+4227795\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^8-195886323*z1^7+66862152*z1^3+691200*z 1-52206228*z1^4)*u[i-3]+(122482002112\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^10+349133680192*z1^9+24107954304*z1^11-56092360704*z1^6+ 643341312*z1^12+2504\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9079795 2*z1^8-125752729600*z1^7)*u[i-9]+(359723655168*z1^11+127806744576*z1^1 0+1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n562886144*z1^13+75496863 744*z1^12-205857423360*z1^9)*u[i-12]+(9583704*z1^2-29647\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n512*z1^5-88820010*z1^6+137781*z1^8-122076 72*z1^7+30524904*z1^3+2903040*z1+813926\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n88*z1^4)*u[i-2]+(36737435016*z1^10+111274290176*z1^ 9-63443050496*z1^6+2963932800\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n*z1^11-16113532928*z1^5+186915449496*z1^8-130636800*z1^12+7908624908 8*z1^7)*u[i-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n8]+(-2133899345 92*z1^8-18649129984*z1^9+19039847424*z1^12+526364157952*z1^10+233\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n259145216*z1^11+1273577472*z1^13 )*u[i-11]+(196830*z1^7-12450915*z1^5-653508*z1^2\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+6613326*z1^4-6626853*z1^6+10000422*z1^3+1402920* z1)*u[i-1]+(-56623104*z1^13*u[i\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-16]+(-349175808*z1^13-121110528*z1^12)*u[i-15]+(-1936588800*z1^ 12-506363904*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n13+60479897 6*z1^11)*u[i-14]+(2795520*z1^2+21543504*z1^4+2776374*z1^8-8414208*z1^ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3-928908*z1^9+10175382*z1^6 -45421878*z1^7+120052452*z1^5)*u[i-4]+(-2250778316*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^8-366192128*z1^6-276616696*z1^10-152173504 4*z1^9-2075002112*z1^7+19143552*z1^11\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+161349632*z1^4+860999680*z1^5)*u[i-7]+(-15017670272*z1^9-27 5730432*z1^12-884736\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^13+ 3859357696*z1^8-11883009536*z1^10-2379926400*z1^11+4439867392*z1^7)*u[ i-1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0]+(3040772096*z1^10-227 082240*z1^13-4176101376*z1^11-3448061952*z1^12)*u[i-13]+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n(50463*z1^8-1617408*z1^2+9468063*z1^7+672 7932*z1^4-4670208*z1^3+5862708*z1^6-921\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n60*z1+375192*z1^5)*u[i-3]+(-24914656*z1^6-823377372 *z1^8-334708408*z1^9+25460736\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n*z1^3+19786200*z1^10+216506368*z1^4-236184476*z1^7+64343808*z1^5)*u[ i-6]+(-12070\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n156864*z1^9-218 97216*z1^12-4478771776*z1^10-8780641792*z1^8+2020540416*z1^6-8810\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n61760*z1^11+4136263680*z1^7)*u[i -9]+(-1991755776*z1^12+5912788992*z1^9-385063936\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^10-9909526528*z1^11-49545216*z1^13)*u[i-12]+ (-15309*z1^8-1006560*z1^2+11284\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n11*z1^7-6286896*z1^4-1934388*z1^3+6670161*z1^6-259200*z1+1738044 *z1^5)*u[i-2]+(-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n203109890*z 1^8-185914218*z1^7+6962688*z1^4+5162304*z1^9-467208*z1^10+35201024*z1 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^3+1474560*z1^2+290210900*z 1^6+151442560*z1^5)*u[i-5]+(-7080728456*z1^8-44922089\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n60*z1^9+5806080*z1^12+2351345664*z1^6-29241 75360*z1^7-117120128*z1^11-1531296024\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^10+658374656*z1^5)*u[i-8]+(135682048*z1^9-607199232*z1^1 2-37158912*z1^13-157\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1251507 2*z1^10-6734263296*z1^11+6481641472*z1^8)*u[i-11]+(-712314*z1^4+103356 *z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^2+698625*z1^6-24057*z1^ 7-1045386*z1^3-129600*z1+1334637*z1^5)*u[i-1])*i)*i)*i)/\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n(-58320-239112*z1+457812*z1^3+52488*z1^6- 494262*z1^5-418446*z1^4+148716*z1^2+(43\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n8372*z1-96228*z1^6+906147*z1^5+106920-839322*z1^3-2 72646*z1^2+767151*z1^4+(-5832\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n0-239112*z1+457812*z1^3+52488*z1^6-494262*z1^5-418446*z1^4+148716*z1 ^2+(-8748*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 84 "\n^6+82377*z1^5+6 9741*z1^4-76302*z1^3-24786*z1^2+39852*z1+9720)*i)*i)*i))) end do; [\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 33 "\nseq(u[i], i = 0 .. n)] end pro c;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I=proc_q_no nRooted_blackVertexG6\"f*6#I\"nGF%6$I\"iGF%I\"uGF%F%F%C4>&8%6#\"\"!F1> &F/6#\"\"\"F1>&F/6#\"\"#F1>&F/6#\"\"$F1>&F/6#\"\"%,$*$I#z1GF%F9\"\")>& 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\"FC!(gl+\"F`p\"(6%G6FX!('*oG'FK!()QM>Fgo\"(h,n'FD!'+#f#Fin\"(W!Q2'Ffv!)7*er$F[u!,s]^7d\"Fgu!+'HjUt'Faq\"+s9k\" ['F5FiyF5F5*&,0FX!'9BrFC\"'cL5Fgo\"'D')pF`p!&dS#FK!('QX5FD!'+'H\"Fin\" (PYL\"F5F^`lF5F5F5FjrF5F5F5FjrF5F5F5FjrF5F5F5,2!&?$eF5FD!'7\"R#FK\"'7y XFgo\"&)[_Fin!'iU\\FX!'Y%=%FC\"';([\"*&,2FD\"'s$Q%Fgo!&Gi*Fin\"'Zh!*\" '?p5F5FK!'A$R)FC!'YEFFX\"'^rw*&,2F\\imF5FDF]imFKF^imFgoF_imFinF`imFXFa imFCFbim*&,0Fgo!%[()Fin\"&xB)FX\"&T(pFK!&-j(FC!&'yCFD\"&_)R\"%?(*F5F5F jrF5F5F5FjrF5F5F5FjrF5F5Fa`lFa`l7#-I$seqGF^s6$F`s/Fjr;F1F\\sF%F%F%" } {TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 32 "proc_q_no nRooted_blackVertex(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 101 "dev_q_nonRooted_blackVertex:=series(expand(subs( z1=z1/z2,t=t*z2,sum(%[i]*t**(i-1),i=1..N+1))),t,N+1):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 35 "The series q_rootedFace_whiteVertex" } {TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 71 "dev_q_root ed_whiteVertex:=subs(\{z1=z2,z2=z1\},dev_q_rooted_blackVertex):" }}}} {SECT 0 {PARA 284 "" 0 "" {TEXT -1 38 "The series q_nonRootedFace_whit eVertex" }{TEXT -1 0 "" }}{EXCHG {PARA 287 "> " 0 "" {MPLTEXT 1 -1 77 "dev_q_nonRooted_whiteVertex:=subs(\{z1=z2,z2=z1\},dev_q_nonRooted_bla ckVertex):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 37 "The series q_no nRootedFace-rootedFace" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 81 "proc_q_nonRooted_rooted := proc (n) local i, u; u[0] \+ := 0; u[1] := 0; u[2] := z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n ; u[3] := 0; u[4] := 4*z1^2; u[5] := 3*z1^3+3*z1^2; u[6] := 25*z1^3; u [7] := 36*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^4+36*z1^3; u[8 ] := 15*z1^5+214*z1^4+15*z1^3; u[9] := 405*z1^5+405*z1^4; u[10]\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n := 319*z1^6+2138*z1^5+319*z1^4; u[11] := 91*z1^4+91*z1^7+4718*z1^6+4718*z1^5; u\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[12] := 5187*z1^7+23404*z1^6+5187*z1^5; u[13] := \+ 2828*z1^5+2828*z1^8+57085*z1^7+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n57085*z1^6; u[14] := 612*z1^9+77660*z1^8+272361*z1^7+77660*z1^6+ 612*z1^5; u[15] \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n:= 60384*z1 ^6+60384*z1^9+713100*z1^8+713100*z1^7; for i from 16 to n do u[i] := \+ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 96 "\nexpand(factor(-(-4041031680 0*z1^10*u[i-15]+(-6181482960*z1^6-214635600*z1^2-2581793820*z1^7-68\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n89643010*z1^5-2581793820*z1^3- 6181482960*z1^4-214635600*z1^8)*u[i-5]+(-936785326\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n188*z1^6-936785326188*z1^7-310487872008*z1^5-56 5678080*z1^3-27783602688*z1^4-565\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n678080*z1^10-310487872008*z1^8-27783602688*z1^9)*u[i-8]+(-246671 0094336*z1^8-131\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0130741120* z1^7-1310130741120*z1^9-154015395840*z1^10-154015395840*z1^6)*u[i-11] \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+(-182452377600*z1^10-18245 2377600*z1^9)*u[i-14]+(149183100*z1^6-787320*z1-84362\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n580*z1^2-84362580*z1^7+1987855020*z1^5+1491 83100*z1^3+1987855020*z1^4-787320*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n8)*u[i-4]+(-402572291286*z1^6-241870438896*z1^7-241870438896 *z1^5-2954441088*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3-47490 900576*z1^4-47490900576*z1^8-2954441088*z1^9)*u[i-7]+(-664643207040*z1 ^9-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n39933941760*z1^5-3993394 1760*z1^10-2233811744760*z1^7-2233811744760*z1^8-6646432\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n07040*z1^6)*u[i-10]+(-331926312960*z1^8-3 31926312960*z1^10-797675222016*z1^9)*u[\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\ni-13]+(53957502*z1^6+1846800*z1+53957502*z1^2+18468 00*z1^7+412546932*z1^5+412546\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n932*z1^3+716610240*z1^4)*u[i-3]+(-82955782398*z1^6-103466880*z1^2-32 674039368*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^7-82955782398* z1^5-4765640328*z1^3-32674039368*z1^4-4765640328*z1^8-103466880*z\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^9)*u[i-6]+(-1121516393040*z1^6 -1957630276740*z1^7-183014542080*z1^5-5455902720\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^4-5455902720*z1^10-1121516393040*z1^8-1830145 42080*z1^9)*u[i-9]+(-1418792292\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n096*z1^8-308855930880*z1^10-1418792292096*z1^9-308855930880*z1^7 )*u[i-12]+(35269\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n02*z1^6+595 16532*z1^3+3526902*z1+59516532*z1^4+28556874*z1^2+28556874*z1^5)*u[i- \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2]+(892782*z1+1405512*z1^2+ 892782*z1^5+1069200*z1^3+1405512*z1^4)*u[i-1]+(911263\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3344*z1^10*u[i-15]+(1743239796*z1^7+1743239 796*z1^3+4952281551*z1^5+4301183448*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^4+4301183448*z1^6+159330816*z1^8+159330816*z1^2)*u[i-5]+(1 1070000128*z1^4+2169\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n81504*z 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"\n34145037*z1^2-299119068*z1^5-1282068*z1^7-34145037*z1^6)*u[i-3]+ (17147924864*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4+262554632 0*z1^3+17147924864*z1^7+41819739407*z1^5+65528064*z1^2+65528064*z1^9+ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n41819739407*z1^6+2625546320 *z1^8)*u[i-6]+(1891995648*z1^4+1891995648*z1^10+66120\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2758748*z1^7+378540800000*z1^6+62262692352* z1^9+62262692352*z1^5+378540800000*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^8)*u[i-9]+(374491445760*z1^8+80363765760*z1^10+374491445760 *z1^9+80363765760*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^7)*u[i -12]+(-60967242*z1^3-30011715*z1^5-3958713*z1-30011715*z1^2-3958713*z1 ^6-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n60967242*z1^4)*u[i-2]+(3 69643413248*z1^7+42768752640*z1^6+700168049984*z1^8+3696\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n43413248*z1^9+42768752640*z1^10)*u[i-11]+ (-13122-1792368*z1^3-2494638*z1^4-16225\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n11*z1-2494638*z1^2-1622511*z1^5-13122*z1^6)*u[i-1]+ (-684195840*z1^10*u[i-15]+(-3\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n87695808*z1^7-387695808*z1^3-1134697500*z1^5-37910016*z1^2-965999817 *z1^4-379100\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n16*z1^8-9659998 17*z1^6)*u[i-5]+(-3367600128*z1^10-3367600128*z1^9)*u[i-14]+(-466\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n560*z1^8-27684000*z1^7-27684000* z1^2-46012950*z1^3-466560*z1+141611859*z1^4-4601\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2950*z1^6+141611859*z1^5)*u[i-4]+(-3246703104*z1^ 4-15056168592*z1^7-24543023250*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^6-215218176*z1^3-215218176*z1^9-15056168592*z1^5-3246703104*z 1^8)*u[i-7]+(-70\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n680062592*z 1^7-20763494400*z1^9-1229291520*z1^5-70680062592*z1^8-20763494400*z1^ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6-1229291520*z1^10)*u[i-10] +(-6736478208*z1^8-16464814080*z1^9-6736478208*z1^10)\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*u[i-13]+(97200*z1+123846408*z1^4+5651208*z 1^2+5651208*z1^6+68203782*z1^5+682037\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n82*z1^3+97200*z1^7)*u[i-3]+(-2997088512*z1^7-479619456*z1^3- 7043943849*z1^5-1314\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2016*z1 ^9-2997088512*z1^4-13142016*z1^2-7043943849*z1^6-479619456*z1^8)*u[i-6 ]+(\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-218923008*z1^10-7478018 1216*z1^7-42791533056*z1^8-7089917952*z1^9-218923008*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4-42791533056*z1^6-7089917952*z1^5)*u[i-9 ]+(-6981550080*z1^10-6981550080*z1^7-33\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n026985984*z1^8-33026985984*z1^9)*u[i-12]+(1389231*z 1^6+9875763*z1^5+1389231*z1+1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n9487628*z1^4+9875763*z1^2+19487628*z1^3)*u[i-2]+(-1470554112*z1^4-27 721728*z1^10\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-44409389664*z1 ^6-44409389664*z1^7-27721728*z1^3-1470554112*z1^9-15089980800*z1^\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n5-15089980800*z1^8)*u[i-8]+(-348 71826432*z1^7-66464491008*z1^8-34871826432*z1^9-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3966566400*z1^6-3966566400*z1^10)*u[i-11]+(128231 1*z1^2+19683+871398*z1+817452*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^3+19683*z1^6+1282311*z1^4+871398*z1^5)*u[i-1]+(17104896*z1^10* u[i-15]+(8808038\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4*z1^10+880 80384*z1^9)*u[i-14]+(62208*z1^8-4728285*z1^4+2825424*z1^2+62208*z1+64 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n29816*z1^3+6429816*z1^6+282 5424*z1^7-4728285*z1^5)*u[i-4]+(728132432*z1^7+163573\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n760*z1^8+163573760*z1^4+11124736*z1^3+11124 736*z1^9+1173879713*z1^6+728132432*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^5)*u[i-7]+(710299648*z1^9+2434563584*z1^8+41680896*z1^10+71 0299648*z1^6+2434563\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n584*z1^ 7+41680896*z1^5)*u[i-10]+(185008128*z1^8+456228864*z1^9+185008128*z1^1 0)*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nu[i-13]+(-66933*z1^6-932 7744*z1^4-66933*z1^2+42768*z1^7-4796658*z1^3+42768*z1-47\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n96658*z1^5)*u[i-3]+(396018400*z1^6+847872 *z1^2+174602656*z1^4+847872*z1^9+291235\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n84*z1^3+174602656*z1^7+29123584*z1^8+396018400*z1^5 )*u[i-6]+(1620208384*z1^8+283\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n2576256*z1^7+1620208384*z1^6+8454144*z1^4+8454144*z1^10+270237696*z1 ^9+270237696\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^5)*u[i-9]+( 973246464*z1^8+202506240*z1^7+973246464*z1^9+202506240*z1^10)*u[i\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-12]+(-145800*z1^6-1941570*z1^4- 1004562*z1^2-1941570*z1^3-145800*z1-1004562*z1^5\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n)*u[i-2]+(70565589*z1^4+2937600*z1^2+2937600*z1^8 +28566432*z1^3+70565589*z1^6+28\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n566432*z1^7+83511639*z1^5)*u[i-5]+(644983552*z1^8+65155072*z1^9+ 1874858128*z1^6+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1874858128* z1^7+1179648*z1^10+1179648*z1^3+65155072*z1^4+644983552*z1^5)*u[i-8]+ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n(2110251520*z1^8+1100038144 *z1^9+1100038144*z1^7+122880000*z1^10+122880000*z1^6)\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*u[i-11]+(-6561-6561*z1^6-193185*z1^4-19318 5*z1^2-94284*z1^3-141669*z1-141669*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^5)*u[i-1])*i)*i)*i)/((-13122-30618*z1+17496*z1^2+17496*z1^3 -30618*z1^4-13122*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^5+(459 27*z1-26244*z1^2+19683-26244*z1^3+45927*z1^4+19683*z1^5+(-6561-15309*z 1+8\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 84 "\n748*z1^2-6561*z1^5+8748* z1^3-15309*z1^4)*i)*i)*i))) end do; [seq(u[i], i = 0 .. n)\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 12 "\n] end proc;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I8proc_q_nonRooted_rootedG6\"f*6#I \"nGF%6$I\"iGF%I\"uGF%F%F%C4>&8%6#\"\"!F1>&F/6#\"\"\"F1>&F/6#\"\"#I#z1 GF%>&F/6#\"\"$F1>&F/6#\"\"%,$*$F:F9FB>&F/6#\"\"&,&*$F:F>F>FDF>>&F/6#\" \"',$FJ\"#D>&F/6#\"\"(,&*$F:FB\"#OFJFW>&F/6#\"\"),(*$F:FH\"#:FV\"$9#FJ Fhn>&F/6#\"\"*,&Fgn\"$0%FVF_o>&F/6#\"#5,(*$F:FN\"$>$Fgn\"%Q@FVFfo>&F/6 #\"#6,*FV\"#\"**$F:FTF]pFeo\"%=ZFgnF_p>&F/6#\"#7,(F^p\"%(=&Feo\"&/M#Fg nFep>&F/6#\"#8,*Fgn\"%GG*$F:FenF\\qF^p\"&&3dFeoF^q>&F/6#\"#9,,*$F:F]o \"$7'F]q\"&gw(F^p\"'hBFFeoFfqFgnFeq>&F/6#Fhn,*Feo\"&%QgFdqF\\rF]q\"'+J rF^pF]r?(8$\"#;F59$I%trueGI*protectedGFcr>&F/6#F_r-I'expandGFcr6#-I'fa ctorGI(_syslibGF%6#,$*(,B*&F:Fco&F/6#,&F_rF5!#:F5F5!,+oJ5/%*&,0Feo!+gH [\"='FD!*+cj9#F^p!+?Qz\"e#Fgn!+5Ik*)oFJF[tFVFisF]qFjsF5&F/6#,&F_rF5!\" &F5F5F5*&,2Feo!-)=E`yO*F^pFctFgn!-3?(y[5$FJ!*!3yccFV!,)o-OyF*$F:FcoFet F]qFdtFdqFftF5&F/6#,&F_rF5!\")F5F5F5*&,,F]q!.OV45nY#F^p!.?6uI,J\"FdqF_ uFgt!-SeR:S:FeoF`uF5&F/6#,&F_rF5!#6F5F5F5*&,&Fgt!-+wP_C=FdqFguF5&F/6#, &F_rF5!#9F5F5F5*&,2Feo\"*+J=\\\"F:!'?tyFD!)!eiV)F^pF`vFgn\"+?]&y)>FJF^ vFVFavF]qF_vF5&F/6#,&F_rF5!\"%F5F5F5*&,0Feo!-'G\"HsDSF^p!-'*)Q/(=CFgnF ivFJ!+)3TW&HFV!,w0!4\\ZF]qF[wFdqFjvF5&F/6#,&F_rF5!\"(F5F5F5*&,.Fdq!-Sq ?VYmFgn!,g<%R$*RFgtFcwF^p!.gZu6QB#F]qFdwFeoFbwF5&F/6#,&F_rF5!#5F5F5F5* &,(F]q!-gHJE>LFgtF[xFdq!-;?AvwzF5&F/6#,&F_rF5!#8F5F5F5*&,0Feo\")-v&R&F :\"(+o%=FDFcxF^pFdxFgn\"*Kpa7%FJFexFV\"*S-h;(F5&F/6#,&F_rF5!\"$F5F5F5* &,2Feo!,)R#ybH)FD!*!)oY.\"F^p!,o$RSnKFgnF]yFJ!+G.klZFVF_yF]qF`yFdqF^yF 5&F/6#,&F_rF5!\"'F5F5F5*&,0Feo!.SIR;:7\"F^p!.SnFIw&>Fgn!-!3UX,$=FV!+?F !fX&FgtFjyF]qFgyFdqFiyF5&F/6#,&F_rF5!\"*F5F5F5*&,*F]q!.'4#H#z=9Fgt!-!) 3$f&)3$FdqFazF^pFbzF5&F/6#,&F_rF5!#7F5F5F5*&,.Feo\"(-p_$FJ\")Kl^fF:Fiz FVFjzFD\")uobGFgnF[[lF5&F/6#,&F_rF5!\"#F5F5F5*&,,F:\"'#y#*)FD\"(7bS\"F gnFb[lFJ\"(+#p5FVFc[lF5&F/6#,&F_rF5!\"\"F5F5F5*&,BFas\"+WLj7\"**&,0F^p \"+'zRKu\"FJF^\\lFgn\"+^:G_\\FV\"+[M=,VFeoF`\\lF]q\"*;3Lf\"FDFa\\lF5F] tF5F5*&,2FV\",G,+q5\"Fgt\"*/:)p@FJFe\\lFdqFd\\lFeo\"-?BL6CNF^pFf\\lF]q \"-!oxGE=\"FgnFg\\lF5FhtF5F5*&,&Fgt\",3=jGH%FdqFj\\lF5FhuF5F5*&,2F]q\" (_Z3\"F^p\")'Hdr)FJ\")%>PW%F:F]]lFDF^]lFV!*EI)**)*FeoF_]lFgnF`]lF5FbvF 5F5*&,0FV\",cE5.:#FJ\"+7po$Q\"Feo\"-\"RuVyr\"F^p\"-/oVoU5FdqFd]lFgnFf] lF]qFc]lF5F\\wF5F5*&,.Fgt\",/*f+77FgnFi]lFdq\"-7@S*3.#FeoFj]lF^p\"-)G' QYooF]qF[^lF5FewF5F5*&,(F]q\",Wt0c=)FgtF^^lFdq\"-g`@b$)>F5F]xF5F5*&,0F J!*o!>\"*HF:!(o?G\"FV!*3I%o_FD!)P]9MFgnFb^lF^pFc^lFeoFe^lF5FgxF5F5*&,2 FV\",k[#z9=%FD\")k!Gb'FdqF[_lFeoFj^lF]qFi ^lF5FayF5F5*&,0FV\"+[c*>*=FgtF^_lF^p\"-[(eF?h'Feo\"-++!3ay$Fdq\",_BpiA 'FgnFa_lF]qF`_lF5F[zF5F5*&,*F]q\"-gdW\"\\u$Fgt\",gdwj.)FdqFd_lF^pFe_lF 5FczF5F5*&,.FJ!)Us'4'Fgn!):<,IF:!(8(eRFDFi_lFeoFj_lFVFh_lF5F\\[lF5F5*& ,,F^p\"-[KTV'p$Feo\",SEvoF%F]q\"-%)*\\!o,qFdqF]`lFgtF^`lF5FauF5F5*&,0! &AJ\"F5FJ!(oBz\"FV!(QY\\#F:!(6Di\"FDFd`lFgnFe`lFeoFb`lF5Fe[lF5F5*&,BFa s!*Se>%o*&,0F^p!*3ep(QFJF[alFgn!++vpM6FD!);+\"z$FV!*<)**f'*F]qF]alFeoF ^alF5F]tF5F5*&,&Fgt!+G,gnLFdqFaalF5FhuF5F5*&,2F]q!'glYF^p!)+SoFFDFealF J!)]H,YF:FdalFV\"*f=hT\"FeoFfalFgnFgalF5FbvF5F5*&,0FV!+/JqYKF^p!,#foh0 :Feo!,]K-VX#FJ!*w\"=_@FdqF]blFgnF[blF]qFjalF5F\\wF5F5*&,.F^p!,#fi+oqFd q!,+W\\j2#Fgn!+?:HH7F]qF`blFeoFablFgtFbblF5FewF5F5*&,(F]q!+3#ykt'Fdq!, !39[Y;FgtFeblF5F]xF5F5*&,0F:\"&+s*FV\"*3k%Q7FD\"(37l&FeoF[clFgn\")#y.# oFJF\\clF^pFiblF5FgxF5F5*&,2F^p!+7&)3(*HFJ!*c%>'z%Fgn!+\\Q%R/(Fdq!);?9 8FVF_clFDFbclFeoFaclF]qF`clF5FayF5F5*&,0Fgt!*3I#*=#F^p!,;7=!yuF]q!,cI` \"zUFdq!+_z\"**3(FVFeclFeoFgclFgnFhclF5F[zF5F5*&,*Fgt!+!3]:)pF^pF[dlF] q!,%)f)p-LFdqF\\dlF5FczF5F5*&,.Feo\"(J#*Q\"Fgn\"(jd()*F:F_dlFV\")Gw[>F DF`dlFJFadlF5F\\[lF5F5*&,2FV!+7Tbq9Fgt!)GF5F:\"')Rr)FJ\"'_u\")Feo F`elFVF_elFgnFaelF5Fe[lF5F5*&,@Fas\")'*[5<*&,&Fgt\")%Q!3))FdqFhelF5Fhu F5F5*&,2F]q\"&3A'FV!(&GGZFD\"(Ca#GF:F[flFJ\"(;)HkFeoF^flF^pF]flFgnF\\f lF5FbvF5F5*&,0F^p\"*KC8G(F]q\"*gPdj\"FVFbflFJ\")OZ76FdqFcflFeo\"+8(zQ< \"FgnFaflF5F\\wF5F5*&,.Fdq\"*['*H5(F]q\"+%ejXV#Fgt\")'*3oTFeoFgflF^pFh flFgnFiflF5FewF5F5*&,(F]q\"*G\"3]=Fdq\"*k)GiXFgtF\\glF5F]xF5F5*&,0Feo! &Lp'FV!(WxK*FDF`glF^p\"&oF%FJ!(emz%F:FbglFgnFcglF5FgxF5F5*&,2Feo\"*+%= gRFD\"'sy%)FV\"*cEgu\"FdqFgglFJ\")%eB\"HF^pFhglF]qFiglFgnFfglF5FayF5F5 *&,0F]q\"+%Q3-i\"F^p\"+cidKGFeoF\\hlFV\"(WTX)FgtF^hlFdq\"*'pP-FFgnF_hl F5F[zF5F5*&,*F]q\"*kkCt*F^p\"*Si]-#FdqFbhlFgtFchlF5FczF5F5*&,.Feo!'+e9 FV!(q:%>FD!(iX+\"FJFghlF:FfhlFgnFhhlF5F\\[lF5F5*&,0FV\")*el0(FD\"(+w$H F]qF\\ilFJ\")KkcGFeoF[ilF^pF]ilFgn\")R;^$)F5F]tF5F5*&,2F]q\"*_N)\\kFdq \")s]:lFeo\"+G\"e[(=F^pFcilFgt\"(['z6FJFdilFVFbilFgnFailF5FhtF5F5*&,,F ]q\"+?:D5@Fdq\"+W\"Q+5\"F^pFhilFgt\"*++)G7FeoFiilF5FauF5F5*&,0!%hlF5Fe oF\\jlFV!'&=$>FDF]jlFJ!&%G%*F:!'p;9FgnF_jlF5Fe[lF5F5F5F_rF5F5F5F_rF5F5 F5F_rF5F5F5,0Fb`lF5F:!&=1$FD\"&'\\ " 0 "" {MPLTEXT 1 0 27 "proc_ q_nonRooted_rooted(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 287 "> " 0 "" {MPLTEXT 1 -1 96 "dev_q_nonRooted_rooted:=series(expand(subs(z1=z1/ z2,t=t*z2,sum(%[i]*t**(i-1),i=1..N+1))),t,N+1):" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 40 "The series q_nonRootedFace_nonRootedFace" } {TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 81 "proc_nonRo oted_nonRooted := proc (n) local i, u; u[0] := 0; u[1] := 0; u[2] := 2 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1; u[3] := 0; u[4] := 16* z1^2; u[5] := 15*z1^3+15*z1^2; u[6] := 150*z1^3; u[7]\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n := 252*z1^4+252*z1^3; u[8] := 120*z1^5+171 2*z1^4+120*z1^3; u[9] := 3645*z1^5+36\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n45*z1^4; u[10] := 3190*z1^6+21380*z1^5+3190*z1^4; u[11] := 1 001*z1^7+51898*z1^6+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n51898*z 1^5+1001*z1^4; u[12] := 62244*z1^7+280848*z1^6+62244*z1^5; u[13] := 36 764\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^8+742105*z1^7+742105 *z1^6+36764*z1^5; u[14] := 8568*z1^9+1087240*z1^8+381305\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4*z1^7+1087240*z1^6+8568*z1^5; u[15] := 9 05760*z1^9+10696500*z1^8+10696500*z1^7+\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n905760*z1^6; u[16] := 403104*z1^10+18034416*z1^9+52 996000*z1^8+18034416*z1^7+403\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n104*z1^6; u[17] := 74613*z1^11+18859647*z1^10+155439585*z1^9+1554395 85*z1^8+1885\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9647*z1^7+74613 *z1^6; u[18] := 12019590*z1^11+291021822*z1^10+749583360*z1^9+291\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 96 "\n021822*z1^8+12019590*z1^7; for i from 19 to n do u[i] := expand(factor(-(7112215756800*z1^13*u\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[i-19]+(-541040640*z1^3+44268903 376720*z1^6+8134819846080*z1^5+224814530434720*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^8+146278246149960*z1^9+146278246149960*z1^7+413 434986240*z1^4+8134819846080*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^11+44268903376720*z1^10-541040640*z1^13+413434986240*z1^12)*u[i -9]+(35188573593\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n60*z1^6+616 486577607328*z1^8+1702744267679464*z1^9+76767499198720*z1^7+616486577 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n607328*z1^11+17027442676794 64*z1^10+3518857359360*z1^13+76767499198720*z1^12)*u[\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\ni-12]+(80753268609024*z1^13+80753268609024* z1^9+643453722921472*z1^12+1198671634\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n164736*z1^11+643453722921472*z1^10)*u[i-15]+(38913510113280* z1^13+38913510113280\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^12) *u[i-18]+(6765669120*z1^3+18135437257482*z1^6+8115058562784*z1^5+27763 628\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n795754*z1^8+181354372574 82*z1^9+27763628795754*z1^7+1202056419552*z1^4+120205641\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9552*z1^11+8115058562784*z1^10+6765669120 *z1^12)*u[i-8]+(19114561194240*z1^6+732\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n566876160*z1^5+822946993376496*z1^8+133272264265375 2*z1^9+191305197631200*z1^7+1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n91305197631200*z1^11+822946993376496*z1^10+732566876160*z1^13+191145 61194240*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n12)*u[i-11]+(37 144169287680*z1^8+495805156942080*z1^9+37144169287680*z1^13+15823\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n64014032000*z1^11+15823640140320 00*z1^10+495805156942080*z1^12)*u[i-14]+(2146489\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n46565120*z1^12+89005280440320*z1^13+8900528044032 0*z1^11)*u[i-17]+(72386388768*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^3-5197632434736*z1^6+2966609994324*z1^5-5197632434736*z1^8+296 6609994324*z1^9-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n13612850106 228*z1^7+1574187659664*z1^4-1289260800*z1^2+72386388768*z1^11+1574187 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n659664*z1^10-1289260800*z1^ 12)*u[i-7]+(41909483991456*z1^6+3862716279552*z1^5+60\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1210944703332*z1^8+601210944703332*z1^9+238 093504256304*z1^7+51266433024*z1^4+41\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n909483991456*z1^11+238093504256304*z1^10+51266433024*z1^13+3 862716279552*z1^12)*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nu[i-10] +(238941229866240*z1^8+1255988233858992*z1^9+12257986129920*z1^7+12559 882\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n33858992*z1^11+212944483 4639904*z1^10+12257986129920*z1^13+238941229866240*z1^12\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n)*u[i-13]+(110232973578240*z1^13+50121495 8069760*z1^12+501214958069760*z1^11+110\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n232973578240*z1^10)*u[i-16]+(144425386256*z1^3-7346 785739018*z1^6-973127221394*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n1^5-973127221394*z1^8+896280904322*z1^9-7346785739018*z1^7+896280904 322*z1^4+518\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n229600*z1^2+518 229600*z1^11+144425386256*z1^10)*u[i-6]+(-2296350-347722794*z1-14\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n30960904*z1^3-1430960904*z1^6+12 453183810*z1^5-347722794*z1^8-2296350*z1^9-28630\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n12878*z1^7+12453183810*z1^4-2863012878*z1^2)*u[i- 2]+(56019600*z1+84125753568*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3-2070753492684*z1^6-966250242720*z1^5+120009602868*z1^8+8412575 3568*z1^9-966250\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n242720*z1^7 +120009602868*z1^4+6568449552*z1^2+56019600*z1^11+6568449552*z1^10)*u \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[i-5]+(-4592700-149293368*z 1+443304900*z1^3-288051228*z1^6+443304900*z1^5-459270\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^8-149293368*z1^7+1185645600*z1^4-28805 1228*z1^2)*u[i-1]+(214575480*z1-12477\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n728130*z1^3-79187837460*z1^6-79187837460*z1^5-12477728130*z1 ^8+3062338470*z1^9-3\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2191928 700*z1^7-32191928700*z1^4+3062338470*z1^2+214575480*z1^10)*u[i-4]+(106 839\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n00*z1-18334197396*z1^3+2 0492537220*z1^6+69982227720*z1^5-4632221736*z1^8+1068390\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^9-18334197396*z1^7+20492537220*z1^4- 4632221736*z1^2)*u[i-3]+(-1232578805760\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n*z1^13*u[i-19]+(183975936*z1^3-14803388013552*z1^6- 2845616990144*z1^5-6848680239\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n2678*z1^8-45440088990886*z1^9-45440088990886*z1^7-143751362048*z1^4- 284561699014\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4*z1^11-1480338 8013552*z1^10+183975936*z1^13-143751362048*z1^12)*u[i-9]+(-694187\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1144960*z1^13-6941871144960*z1^1 2)*u[i-18]+(-2868027904*z1^3-6825915490536*z1^6-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3306156853808*z1^5-8860905055977*z1^8-68259154905 36*z1^9-8860905055977*z1^7-4928\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n64916608*z1^4-492864916608*z1^11-3306156853808*z1^10-2868027904* z1^12)*u[i-8]+(-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n39846950862 848*z1^12-16388297990144*z1^11-16388297990144*z1^13)*u[i-17]+(-386641 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n81376*z1^3+1931980588192*z1 ^6-1548805528006*z1^5+1931980588192*z1^8-154880552800\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6*z1^9+5660667157656*z1^7-754378445296*z1^4 +649059840*z1^2-38664181376*z1^11-754\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n378445296*z1^10+649059840*z1^12)*u[i-7]+(-12730123003264*z1^ 6-1182350852608*z1^5\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-170312 222910242*z1^8-170312222910242*z1^9-68734254249344*z1^7-15101485056*z1 ^4-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n12730123003264*z1^11-687 34254249344*z1^10-15101485056*z1^13-1182350852608*z1^12)\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*u[i-10]+(-97047738513408*z1^11-970477385 13408*z1^12-21044337647616*z1^10-210443\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n37647616*z1^13)*u[i-16]+(-88668610432*z1^3+33010460 91344*z1^6+281245911845*z1^5+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n281245911845*z1^8-516324080076*z1^9+3301046091344*z1^7-516324080076* z1^4-8444375\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n04*z1^2-8444375 04*z1^11-88668610432*z1^10)*u[i-6]+(-416869788053576*z1^9-8988192\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n52224*z1^6-19193033174528*z1^7-1 9193033174528*z1^12-151136406243200*z1^8-1511364\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n06243200*z1^11-898819252224*z1^13-416869788053576 *z1^10)*u[i-12]+(-1612809783705\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6*z1^9-130453494555648*z1^10-16128097837056*z1^13-13045349455564 8*z1^12-24447718\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0990464*z1^ 11)*u[i-15]+(-51438240*z1-65789967826*z1^3+1112650423804*z1^6+4885653 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n42766*z1^5-109329830426*z1^ 8-65789967826*z1^9+488565342766*z1^7-109329830426*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n4-5394231696*z1^2-51438240*z1^11-5394231696 *z1^10)*u[i-5]+(-5311000867328*z1^6-1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n98808150016*z1^5-216663114610448*z1^8-350126992717896*z1^9-5 1327684150912*z1^7-5\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1327684 150912*z1^11-216663114610448*z1^10-198808150016*z1^13-5311000867328*z1 ^12\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n)*u[i-11]+(-790973511270 4*z1^8-340891116443264*z1^10-106003534345728*z1^12-10600\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3534345728*z1^9-340891116443264*z1^11-790 9735112704*z1^13)*u[i-14]+(-175375368*z\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n1-4212257775*z1^3+72666756822*z1^6+72666756822*z1^5 -4212257775*z1^8-3509259408*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n1^9+14202489702*z1^7+14202489702*z1^4-3509259408*z1^2-175375368*z1^1 0)*u[i-4]+(-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n287221597815968 *z1^9-2862265835520*z1^7-54616847471104*z1^8-54616847471104*z1^12\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-287221597815968*z1^11-488254727 376768*z1^10-2862265835520*z1^13)*u[i-13]+(24567\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n462*z1+9816987672*z1^3-10801717308*z1^6-389091716 10*z1^5+2339526132*z1^8+2456746\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2*z1^9+9816987672*z1^7-10801717308*z1^4+2339526132*z1^2)*u[i-3]+ (3521070+3113099\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n73*z1+10249 86393*z1^3+1024986393*z1^6-9554141565*z1^5+311309973*z1^8+3521070*z1^ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n9+2184154956*z1^7-955414156 5*z1^4+2184154956*z1^2)*u[i-2]+(5152572+163348488*z1-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n486827658*z1^3+316385028*z1^6-486827658*z1^ 5+5152572*z1^8+163348488*z1^7-1287977\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n760*z1^4+316385028*z1^2)*u[i-1]+(71156367360*z1^13*u[i-19]+( 412900392960*z1^13+4\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1290039 2960*z1^12)*u[i-18]+(2468924882944*z1^12+1006727790592*z1^11+100672779 059\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2*z1^13)*u[i-17]+(254569 503842*z1^5-106844160*z1^2+6649033216*z1^3-268677816197*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^8+119192328512*z1^4-841784093826*z1^7- 268677816197*z1^6+254569503842*z1^9+119\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n192328512*z1^10-106844160*z1^12+6649033216*z1^11)*u [i-7]+(6276651696128*z1^12+13\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n40876455936*z1^13+6276651696128*z1^11+1340876455936*z1^10)*u[i-16]+( 96510309766*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^9+268723200* z1^2+17602961248*z1^3-23560509922*z1^8+17602961248*z1^10-23560509\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n922*z1^5-519136762918*z1^6+26872 3200*z1^11-519136762918*z1^7+96510309766*z1^4)*u\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n[i-6]+(330963489792*z1^11-20840448*z1^13+16373551 104*z1^12+1643289698816*z1^10+6\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n863530052504*z1^8+1643289698816*z1^6+16373551104*z1^4+4660450957 218*z1^9+4660450\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n957218*z1^7 -20840448*z1^3+330963489792*z1^5)*u[i-9]+(8838352461824*z1^10+1075616 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n219136*z1^9+8838352461824*z 1^12+1075616219136*z1^13+16669154287616*z1^11)*u[i-15\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n]+(-208439057954*z1^6-87148585436*z1^5+1404 433536*z1^2+27468157693*z1^8+15950100\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n758*z1^9+1404433536*z1^10+15240960*z1^11+15950100758*z1^3+27 468157693*z1^4-87148\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n585436* z1^7+15240960*z1)*u[i-5]+(350613504*z1^3+66693336064*z1^4+836354038062 *z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n^6+445073757920*z1^10+86 0877025254*z1^8+66693336064*z1^11+836354038062*z1^9+4450\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n73757920*z1^5+350613504*z1^12+86087702525 4*z1^7)*u[i-8]+(24559711491072*z1^11+56\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n3060867072*z1^8+24559711491072*z1^10+7574322663424* z1^12+563060867072*z1^13+7574\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n322663424*z1^9)*u[i-14]+(-21286216836*z1^5+3488776290*z1^8+118123439 4*z1^9-21286\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n216836*z1^6+118 1234394*z1^2+3488776290*z1^3+47555424*z1^10-2028184956*z1^7-20281\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n84956*z1^4+47555424*z1)*u[i-4]+( 1286548320256*z1^11+16049997890244*z1^8+66043682\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n17760*z1^7+120297058304*z1^12+6604368217760*z1^10 +120297058304*z1^5+128654832025\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n6*z1^6+1478557696*z1^13+16049997890244*z1^9+1478557696*z1^4)*u[i -10]+(4170753716\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n224*z1^8+22 3404032000*z1^7+37445119201280*z1^10+4170753716224*z1^12+223404032000 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n*z1^13+21960847172864*z1^9+ 21960847172864*z1^11)*u[i-13]+(1536784758*z1^6-269063\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n235*z1^8-1512399330*z1^3-7150842*z1^9+63362 02590*z1^5-269063235*z1^2-1512399330*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^7+1536784758*z1^4-7150842*z1)*u[i-3]+(12375642672128*z1^1 1+1600966529024*z1^7\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n+123756 42672128*z1^8+76543229952*z1^13+34116293369792*z1^9+34116293369792*z1^ 10+\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1600966529024*z1^12+7654 3229952*z1^6)*u[i-12]+(-529943526*z1^2-230737788*z1^6-13\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n77810-89169498*z1^8-230737788*z1^3+234613 9350*z1^5-1377810*z1^9-89169498*z1+2346\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n139350*z1^4-529943526*z1^7)*u[i-2]+(4591159223808*z 1^11+4591159223808*z1^7+30707\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n231324896*z1^9+19036615685184*z1^10+19036615685184*z1^8+17975934976* z1^13+491191\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n185408*z1^12+17 975934976*z1^5+491191185408*z1^6)*u[i-11]+(171723240*z1^5-1116271\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n53*z1^6-111627153*z1^2-1869885-1 869885*z1^8+171723240*z1^3+447557400*z1^4-573300\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n18*z1-57330018*z1^7)*u[i-1]+((-51060801536*z1^12- 20632567808*z1^11-20632567808*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n1^13)*u[i-17]+(-28514975744*z1^13-135602470912*z1^12-13560247091 2*z1^11-28514975\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n744*z1^10)* u[i-16]-1368391680*z1^13*u[i-19]+(-23956291584*z1^13-379967152128*z1^ \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n11-200100814848*z1^12-23956 291584*z1^9-200100814848*z1^10)*u[i-15]+(-8188723200*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^13-8188723200*z1^12)*u[i-18]+(-134026362 88*z1^8-13402636288*z1^13-59172158771\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 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"\n*z1^6-933076765312*z1^10-44554633216*z1^7 -933076765312*z1^9-44554633216*z1^12)*u\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n[i-12]+(153090+40001472*z1^7+8011791*z1^8-181704195 *z1^4+40001472*z1^2+8011791*z\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 " \n1+153090*z1^9-181704195*z1^5+16023771*z1^3+16023771*z1^6)*u[i-2]+(55 02665050*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n5+5502665050*z1 ^7-2130022325*z1^8-116382432*z1^2-1451520*z1^11-2130022325*z1^4-1\\" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n16382432*z1^10+13567889134*z1^6- 1244111920*z1^9-1451520*z1-1244111920*z1^3)*u[i-\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n5]+(-33495540240*z1^6-23309331255*z1^8-2330933125 5*z1^7-2994305536*z1^11-1217331\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n2*z1^12-19916092128*z1^10-19916092128*z1^5-33495540240*z1^9-2994 305536*z1^4-1217\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n3312*z1^3)* u[i-8]+(-15126495232*z1^6-136891808256*z1^7-15126495232*z1^12-5578344 \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n35872*z1^8-557834435872*z1^ 10-136891808256*z1^11-898558948048*z1^9-541589504*z1^\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n13-541589504*z1^5)*u[i-11]+(12723723*z1^6+6 489558*z1+6489558*z1^7+220887-4998996\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n0*z1^4-19567818*z1^5+12723723*z1^2-19567818*z1^3+220887*z1^8 )*u[i-1])*i)*i)*i)/(\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n-406782 -8183754*z1+22088700*z1^3-8183754*z1^6-1215972*z1^5-406782*z1^7+220887 00*\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\nz1^4-1215972*z1^2+(-4049 5950*z1^3+15003549*z1+745767*z1^7+745767+2229282*z1^2+22\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 82 "\n29282*z1^5+15003549*z1^6-40495950*z1^4+(- 406782-8183754*z1+22088700*z1^3-8183754\\" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 82 "\n*z1^6-1215972*z1^5-406782*z1^7+22088700*z1^4-121597 2*z1^2+(67797*z1^7+1363959*z1\\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 84 " \n^6+202662*z1^5-3681450*z1^4-3681450*z1^3+202662*z1^2+1363959*z1+6779 7)*i)*i)*i))) \\" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 42 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FjgmFigmF5F5F[tF5F5F5F[tF5F5F5F[tF5F5Fg_lFg_l7#-I$seqGF_t6$Fat/F[t;F1F ]tF%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 28 "proc_nonRooted_nonRooted(N):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 99 "dev_q_nonRooted_nonRooted:=series(expan d(subs(z1=z1/z2,t=t*z2,sum(%[i]*t**(i-1),i=1..N+1))),t,N+1):" }}}} {SECT 0 {PARA 284 "" 0 "" {TEXT -1 39 "The series q of rooted quadrang ulations" }{TEXT -1 0 "" }}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 3383 "proc_q := proc (n) local i, u; u[0] := 0; u[1] := 0; u[2] := 0; \+ u[3] := 0; u[4] := 0; u[5] := 0; u[6] := z1^3; u[7] := 0; u[8] := 4*z1 ^4; u[9] := 3*z1^5+3*z1^4; u[10] := 24*z1^5; u[11] := 33*z1^6+33*z1^5; u[12] := 13*z1^7+188*z1^6+13*z1^5; u[13] := 338*z1^7+338*z1^6; for i \+ from 14 to n do u[i] := expand(factor( -(388147200*z1^7*u[i- 11]+(-13122*z1^4-952560*z1^2-477738*z1-477738*z1^3-13122)*u[i-1]+(3010 96908*z1^4+301096908*z1^3+64791456*z1^2+1760616*z1^6+64791456*z1^5+176 0616*z1)*u[i-4]+(342712320*z1^7+7131941772*z1^5+3664946304*z1^6+366494 6304*z1^4+342712320*z1^3)*u[i-7]+(1434648960*z1^7+1434648960*z1^6)*u[i -10]+(9329454*z1^2+9329454*z1^4+2284524*z1+2284524*z1^5+11877264*z1^3) *u[i-3]+(22550400*z1^2+954271824*z1^3+3810531336*z1^5+22550400*z1^7+95 4271824*z1^6+3810531336*z1^4)*u[i-6]+(4809401280*z1^6+2002467840*z1^5+ 2002467840*z1^7)*u[i-9]+(62208*z1^4-6469308*z1^3-6469308*z1^2+62208*z1 )*u[i-2]+(1713555816*z1^4+894774456*z1^5+72479616*z1^2+72479616*z1^6+8 94774456*z1^3)*u[i-5]+(6156145800*z1^6+6156145800*z1^5+1276128000*z1^7 +1276128000*z1^4)*u[i-8]+((-447136000*z1^7-447136000*z1^6)*u[i-10]+(-6 51525120*z1^5-1589992960*z1^6-651525120*z1^7)*u[i-9]+(-437655040*z1^4- 437655040*z1^7-2167038960*z1^5-2167038960*z1^6)*u[i-8]-116638720*z1^7* u[i-11]+(24057*z1^4+1013472*z1^2+24057+532737*z1+532737*z1^3)*u[i-1]+( -160351830*z1^4-160351830*z1^3-37353984*z1^2-1232496*z1^6-37353984*z1^ 5-1232496*z1)*u[i-4]+(-2730470508*z1^5-1383601216*z1^4-1383601216*z1^6 -126085120*z1^7-126085120*z1^3)*u[i-7]+(-6519006*z1^4-6394095*z1^3-175 3164*z1^5-6519006*z1^2-1753164*z1)*u[i-3]+(-1596899952*z1^5-394313248* z1^6-9457920*z1^7-1596899952*z1^4-9457920*z1^2-394313248*z1^3)*u[i-6]+ (5150439*z1^2+5150439*z1^3-17334*z1^4-17334*z1)*u[i-2]+(-417544104*z1^ 5-36126528*z1^2-804827991*z1^4-36126528*z1^6-417544104*z1^3)*u[i-5]+(( 350073312*z1^5+175056384*z1^6+15482880*z1^7+175056384*z1^4+15482880*z1 ^3)*u[i-7]+(46540800*z1^7+46540800*z1^6)*u[i-10]+(1320960*z1^2+5465318 4*z1^6+224111136*z1^4+1320960*z1^7+54653184*z1^3+224111136*z1^5)*u[i-6 ]+(70901760*z1^5+70901760*z1^7+175825920*z1^6)*u[i-9]+(65380896*z1^3+1 26233916*z1^4+6038016*z1^2+6038016*z1^6+65380896*z1^5)*u[i-5]+(2554464 00*z1^5+50196480*z1^7+255446400*z1^6+50196480*z1^4)*u[i-8]+11673600*z1 ^7*u[i-11]+(-341820*z1^2-191808*z1^3-13122*z1^4-13122-191808*z1)*u[i-1 ]+(7244256*z1^2+28496898*z1^4+28496898*z1^3+7244256*z1^5+279936*z1^6+2 79936*z1)*u[i-4]+(1517454*z1^4+445824*z1^5+1129464*z1^3+445824*z1+1517 454*z1^2)*u[i-3]+(-1335528*z1^3-9882*z1^4-1335528*z1^2-9882*z1)*u[i-2] +((-1694700*z1^4-20736*z1^6-20736*z1-473664*z1^5-473664*z1^2-1694700*z 1^3)*u[i-4]+(-634880*z1^7-15042768*z1^5-7425536*z1^6-634880*z1^3-74255 36*z1^4)*u[i-7]+(-1617920*z1^7-1617920*z1^6)*u[i-10]+(-117846*z1^2-117 846*z1^4-37584*z1^5-37584*z1-68265*z1^3)*u[i-3]+(-61440*z1^2-61440*z1^ 7-2544128*z1^3-10542912*z1^4-10542912*z1^5-2544128*z1^6)*u[i-6]+(-6506 240*z1^6-2580480*z1^7-2580480*z1^5)*u[i-9]+(2916*z1^4+112509*z1^3+1125 09*z1^2+2916*z1)*u[i-2]+(-6618621*z1^4-3443424*z1^5-3443424*z1^3-33868 8*z1^6-338688*z1^2)*u[i-5]+(-10087680*z1^6-1925120*z1^7-1925120*z1^4-1 0087680*z1^5)*u[i-8]-389120*z1^7*u[i-11]+(2187*z1^4+22437*z1^3+36612*z 1^2+22437*z1+2187)*u[i-1])*i)*i)*i)/(-13122-13122*z1^3-16038*z1^2-1603 8*z1+(24057+29403*z1+24057*z1^3+29403*z1^2+(-13122-13122*z1^3-16038*z1 ^2-16038*z1+(2187*z1^3+2673*z1^2+2673*z1+2187)*i)*i)*i))) end do; [seq (u[i], i = 0 .. n)] end proc;" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>I'proc_qG6\"f*6#I\"nGF%6$I\"iGF%I\"uGF%F%F%C2>&8%6#\" \"!F1>&F/6#\"\"\"F1>&F/6#\"\"#F1>&F/6#\"\"$F1>&F/6#\"\"%F1>&F/6#\"\"&F 1>&F/6#\"\"'*$I#z1GF%F=>&F/6#\"\"(F1>&F/6#\"\"),$*$FKFAFA>&F/6#\"\"*,& *$FKFEF=FUF=>&F/6#\"#5,$Fen\"#C>&F/6#\"#6,&*$FKFI\"#LFenFbo>&F/6#\"#7, (*$FKFO\"#8Fao\"$)=FenFio>&F/6#Fio,&Fho\"$Q$FaoF_p?(8$\"#9F59$I%trueGI *protectedGFep>&F/6#Fap-I'expandGFep6#-I'factorGI(_syslibGF%6#,$*&,:*& FKFO&F/6#,&FapF5!#6F5F5\"*+s9)Q*&,,FU!&AJ\"*$FKF9!'gD&*FK!'QxZFJF^rF[r F5F5&F/6#,&FapF5!\"\"F5F5F5*&,.FU\"*3p4,$FJFerF\\r\")c9zkFao\"(;1w\"Fe nFfrFKFgrF5&F/6#,&FapF5!\"%F5F5F5*&,,Fho\"*?BrU$Fen\"+s<%>8(Fao\"+/j% \\m$FUF`sFJF^sF5&F/6#,&FapF5!\"(F5F5F5*&,&Fho\"+g*[YV\"FaoFgsF5&F/6#,& FapF5!#5F5F5F5*&,,F\\r\"(a%H$*FUF^tFK\"(CXG#FenF_tFJ\")ks(=\"F5&F/6#,& FapF5!\"$F5F5F5*&,.F\\r\")+/bAFJ\"*C=Fa*Fen\"+O8`5QFhoFgtFaoFhtFUFitF5 &F/6#,&FapF5!\"'F5F5F5*&,(Fao\"+!G,%4[Fen\"+SyY-?FhoFauF5&F/6#,&FapF5! \"*F5F5F5*&,*FU\"&3A'FJ!(3$pkF\\rFiuFKFhuF5&F/6#,&FapF5!\"#F5F5F5*&,,F U\"+;eb8lFJ!(&4%R'Fen!(kJv\"F\\rF^yFKF`yF5FatF5F5*&,.Fen! +_***of\"Fao!*[KJ%RFho!(?zX*FUFcyF\\rFeyFJFdyF5FjtF5F5*&,*F\\r\"(R/:&F JFhyFU!&Mt\"FKFiyF5FjuF5F5*&,,Fen!*/Ta<%F\\r!)Gl7OFU!*\"*z#[!)FaoF]zFJ F\\zF5FcvF5F5*&,:*&,,Fen\"*7L2]$Fao\"*%Qc]]FaoFi[lFUFj[lF5F[wF5F5Fcq\")+On6*&,,F\\r!'?=MFJ!'3=>FU F[rF[rF5FKF_\\lF5F_rF5F5*&,.F\\r\"(cUC(FU\"))*o\\GFJFc\\lFenFb\\lFao\" 'O*z#FKFd\\lF5FhrF5F5*&,,FU\"(au^\"Fen\"'CeWFJ\"(k%H6FKFh\\lF\\rFg\\lF 5FatF5F5*&,*FJ!(GbL\"FU!%#))*F\\rF\\]lFKF]]lF5FjuF5F5*&,8*&,.FU!(+Zp\" Fao!&O2#FKFc]lFen!'kOZF\\rFd]lFJFb]lF5FhrF5F5*&,,Fho!'!)[jFen!)oF/:Fao !(ObU(FJFg]lFUFi]lF5FasF5F5*&,&Fho!(?zh\"FaoF\\^lF5FhsF5F5*&,,F\\r!'Yy 6FUF_^lFen!&%ePFKF`^lFJ!&l#oF5FatF5F5*&,.F\\r!&S9'FhoFd^lFJ!(GTa#FU!)7 Ha5FenFf^lFaoFe^lF5FjtF5F5*&,(Fao!(Si]'Fho!(![!e#FenFj^lF5FbuF5F5*&,*F U\"%;HFJ\"'4D6F\\rF^_lFKF]_lF5FjuF5F5*&,,FU!(@'=mFen!(CMW$FJFb_lFao!') oQ$F\\rFc_lF5FcvF5F5*&,*Fao!)!o(35Fho!(?^#>FUFg_lFenFf_lF5F[wF5F5Fcq!' ?\"*Q*&,,FU\"%(=#FJ\"&PC#F\\r\"&7m$FKF\\`lF[`lF5F5F_rF5F5F5FapF5F5F5Fa pF5F5F5FapF5F5F5,,F[rF5FJF[rF\\r!&Qg\"FKF_`l*&,,F_xF5FK\"&.%HFJF_xF\\r Fb`l*&,,F[rF5FJF[rF\\rF_`lFKF_`l*&,*FJF[`lF\\r\"%tEFKFg`lF[`lF5F5FapF5 F5F5FapF5F5F5FapF5F5FbrFbr7#-I$seqGFep6$Fgp/Fap;F1FcpF%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 10 "proc_q(N):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 287 "> " 0 "" {MPLTEXT 1 -1 79 "dev_ q:=series(expand(subs(z1=z1/z2,t=t*z2,sum(%[i]*t**(i-1),i=1..N+1))),t, N+1):" }}}}}{SECT 0 {PARA 284 "" 0 "" {TEXT -1 54 "Obtaining the enume ration of unrooted 3-connected maps" }{TEXT -1 0 "" }}{EXCHG {PARA 276 "" 0 "" {TEXT 223 48 "We apply Burnside Formula to get the enumera tion" }{TEXT 223 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 90 "Burnsi de-Liskovets formula can be easily adapted to give the following equat ion on series " }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 683 "sum_k_rooted_3connected_two_variables:=convert(expan d(series(dev_q+z1/z2*subs(z1=z1**2,z2=z2**2,t=t**2,dev_qbb_order2)+z2/ z1*subs(z1=z1**2,z2=z2**2,t=t**2,dev_qww_order2)+subs(z1=z1**2,z2=z2** 2,t=t**2,dev_qbw_order2)+t*z1*subs(z1=z1**2,z2=z2**2,t=t**2,dev_q_root ed_whiteVertex+dev_q_nonRooted_whiteVertex)+t*z2*subs(z1=z1**2,z2=z2** 2,t=t**2,dev_q_rooted_blackVertex+dev_q_nonRooted_blackVertex)+t**2*z1 *z2*subs(z1=z1**2,z2=z2**2,t=t**2,dev_q_nonRooted_rooted+dev_q_nonRoot ed_nonRooted)+sum(phi(k)*(z1/z2*subs(z1=z1**k,z2=z2**k,t=t**k,dev_qbb_ orderNot2)+z2/z1*subs(z1=z1**k,z2=z2**k,t=t**k,dev_qww_orderNot2)+subs (z1=z1**k,z2=z2**k,t=t**k,dev_qbw_orderNot2)),k=3..N),t,N+1)),polynom) :" }}}{EXCHG {PARA 270 "" 0 "" {TEXT 206 192 "By dividing each coeffic ient by 2*(i+j), we get enumeration of unrooted 3-connected maps. The \+ coefficient of z1^i*z2^j is the number of unrooted 3-connected map wit h i+1 vertices and j+1 faces" }{TEXT 206 0 "" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 122 "non_rooted_3connected_map_two_variables:=expan d(factor(series(1/2*int(sum_k_rooted_3connected_two_variables/t,t),t,N +1)));" }{MPLTEXT 1 0 0 "" }}{PARA 273 "" 1 "" {XPPMATH 20 "6#>IHnon_r ooted_3connected_map_two_variablesG6\"+ivI\"tGF%*&I#z1GF%\"\"$I#z2GF%F *\"\"'*&F)\"\"%F+F.\"\"),&*&F)\"\"&F+F.\"\"\"*&F)F.F+F2F3\"\"*,$*&F)F2 F+F2F*\"#5,&*&F+F2F)F,\"\"#*&F)F2F+F,F;\"#6,(*&F)F,F+F,F=*&F)\"\"(F+F2 F;*&F)F2F+FAF;\"#7,&*&F)FAF+F,\"#;*&F)F,F+FAFF\"#8,(*&F)FAF+FA\"#p*&F) F,F+F/F8*&F)F/F+F,F8\"#9,**&F)F,F+F5F,*&F)FAF+F/\"$F\"*&F)F/F+FAFR*&F) F5F+F,F,\"#:,(*&F)FAF+F5\"$G\"*&F)F5F+FAFX*&F)F/F+F/\"$T&FF,**&F)FAF+F 8\"#g*&F)F5F+F/\"%)=\"*&F)F/F+F5Fjn*&F)F8F+FAFhn\"#<,,*&F)FAF+F=F]o*&F )F8F+F/\"%T9*&F)F5F+F5\"%'4&*&F)F/F+F8Fao*&F)F=F+FAF]o\"#=,**&F)F5F+F8 \"&#)>\"*&F)F/F+F=\"%K5*&F)F=F+F/F[p*&F)F8F+F5Fio\"#>,,*&F)F8F+F8\"&'e 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