This Maple session accompanies the paper "THE ISING MODEL ON CUBIC MAPS: ARBITRARY GENUS"Some comments are in French...restart:libname:="/home/bousquet/Calculs/Gfun/gfun", libnamewith(gfun):gfun[version]<Text-field style="Heading 1" layout="Heading 1">Bipartite maps with vertex degrees at most 3: generating the first terms of the series H. Checking the first KP equation (Section 2.2).</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvOn utilise le paquetage Combstruct pour la g\303\251n\303\251ration des partitions et permutationswith(combstruct)Partitions en parts de taille au plus 3Par:={L3=Prod(Z,Z,Z),L2=Prod(Z,Z),L1=Z,AZ=Atom,Pa=Prod(Sequence(L3),Sequence(L2),Sequence(L1))}res:=allstructs([Pa,Par,unlabelled],size=5)Order:=10: gfseries(Par,labelled,z)V\303\251rifionsseries(1/(1-z)/(1-z^2)/(1-z^3),z,10);Traitement du r\303\251sultat pour que la forme soit plus acceptablesimplify_par:=proc(pa) local pa1:
pa1:=eval(subs(Prod=( ()->([args]) ), Sequence=( ()->args ), Epsilon=NULL, pa)):
map(x->nops(x),pa1):
end:map(simplify_par,res);On passe aux permutations \303\240 petits cyclesPer:={A3=Cycle(Z, card=3),A2=Cycle(Z, card=2),A1=Cycle(Z, card=1),Z=Atom,Pe=Set(Union(A1,A2,A3))}res:=allstructs([Pe,Per,labelled],size=4)Order:=10: gfseries(Per,labelled,z)V\303\251rifionsseries(exp(z+z^2/2+z^3/3),z,10)Un paquetage qui va multiplier des permutations, notammentwith(GroupTheory):Transforme une permutation avec \303\251ventuellement des cycles de longueur 1, produite par la proc\303\251dure de g\303\251n\303\251ration, en une permutation sans ces cyclessimplify_perm:=proc(perm) local res,i,res1:
res:=eval(subs(Cycle=( ()->[args] ), Set=( ()->[args ]), perm)):
res:=subs(seq(Z[i]=i,i=1..20),res):
res1:=res:
for i from nops(res1) by -1 to 1 do
if nops(res1[i])=1 then res:=subsop(i=NULL, res) fi:
od: Perm(res):
end:map(simplify_perm, res);
map (PermCycleType,%)nops(%)lambda:=[3,3,2,1,1,1]: convert(map (i->p[i],lambda),`*`);mu:=[3,3,2]: n:=10:q[1]^n*convert(map (i->q[i]/q[1]^i,mu),`*`);
Etant donn\303\251e une partition, on fabrique une permutation canonique ayant ce type cycliqueperm_canonique:=proc(lambda,i) local part,j:
if nops (lambda)=0 then []
else
part:= lambda[1]: if part =1 then []
else
[[seq(i+j,j=0..part-1)],op(perm_canonique(subsop(1=NULL,lambda),i+part))]: fi: fi:
end:
perm_canonique([3,3,3,2,1,1],3);Perm(%);PermCycleType(%)Nombre de permutations de type cyclique lambdaz_lambda:=proc(lambda) local n,m,i,j:
n:=convert(lambda,`+`):
m[1]:=0: m[2]:=0: m[3]:=0:
for i from 1 to nops(lambda) do
j:=lambda[i]:
m[j]:=1+m[j]: od:
#print(m[1],m[2],m[3]);
n!/convert(lambda,`*`)/m[1]!/m[2]!/m[3]!: end:z_lambda([3,3,2,2,2,1,1])14!/9/8/2/6/2Coefficient de z^n/n! dans la s\303\251rie tau des cartes biparties non n\303\251cessairement connexes. Les p_i et q_i comptent les sommets blancs/noirs de degr\303\251 i, et u compte les facesco_tau:=proc(n) local L_set, M_set, mu,res, type, ell, lambda,p_L,sigma_L,sigma_M,q_M,sigma:
L_set:=map(simplify_par,allstructs([Pa,Par,unlabelled],size=n)):
M_set:=map(simplify_perm,allstructs([Pe,Per,labelled],size=n)):
res:=0:
for lambda in L_set do
p_L:=convert(map (i->p[i],lambda),`*`):
sigma_L:=Perm(perm_canonique(lambda,1)):
for sigma_M in M_set do
mu:=PermCycleType(sigma_M):
q_M:=q[1]^n*convert(map (i->q[i]/q[1]^i,mu),`*`):
sigma:=PermProduct(sigma_L,sigma_M):
type:=PermCycleType(sigma):
ell:=nops(type)+n-convert(type,`+`):
res:=res+z_lambda(lambda)*u^(ell)*p_L*q_M:
od:
od:
res: end:co_tau(1);co_tau(2)tau_ser:=proc(N) local n:
1+add(co_tau(n)*z^n/n!,n=1..N): end:tau_ser(3)S\303\251rie pour les cartes connexesH_ser:=proc(N) option remember:
factor(series(log(tau_ser(N)),z,N+1)): end:H_ser(3)expand(coeff(%,z,3)*6)The first KP equation: hurray !n:=8:
normal(series(diff(H_ser(n),p[2]$2)+1/12*diff(H_ser(n),p[1]$4)+1/2*diff(H_ser(n),p[1]$2)^2-diff(diff(H_ser(n),p[1]),p[3]),z,n+1))The outcome of this first section is the series H_ser that counts bipartite maps with small degrees, z recording the edge number, u the face number, and the pi_i, q_i the number of white/black vertices of degree i. This is the series of Section 2H_ser(4)We can go up to 8 edgesH_ser(8):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 1" layout="Heading 1">The series B of bipartite maps with vertex degrees 2 and 3, and its PDE (Section 2.2)</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvThe series for maps with degrees 2 and 3B_ser:=proc(n) subs(p[1]=0, q[1]=0, H_ser(n)):endB_ser(5)One face mapsn:=5: coeff(convert(B_ser(n),polynom),u,1);On aura besoin d'exprimer diff(H,p1$2) et d'autres d\303\251riv\303\251es, sp\303\251cialis\303\251es \303\240 p1=q1=0.
L'op\303\251rateur L (here we include the variables p3 and q3, but they are set to 1 in the paper).opeL:=proc(H)
2*z^2*q[2]*p[3]/(1-z^2*p[2]*q[2])*diff(H,p[2])+ 2*z*q[3]/(1-z^2*p[2]*q[2])*diff(H,q[2]):end:H11H11B:=opeL(opeL(B(z,p[2],q[2])))+u*z^2*q[2]/(1-z^2*p[2]*q[2]):Checkingn:=8: normal(series(subs(B(z,p[2],q[2])=B_ser(n),H11B),z,n+1));
map(normal,subs(p[1]=0,q[1]=0,series (diff(H_ser(n),p[1]$2),z,n+1)));
factor(series(%%-%,z,n+1))H1111H1111B:=opeL(opeL(H11B)):factor(opeL(opeL(u*z^2*q[2]/(1-z^2*p[2]*q[2]))))n:=8: normal(series(subs(B(z,p[2],q[2])=B_ser(n),H1111B),z,n+1));
map(normal,subs(p[1]=0,q[1]=0,series (diff(H_ser(n),p[1]$4),z,n+1)));
factor(series(%%-%,z,n+1))H13H13B:=1/3/p[3]*(z*diff(opeL(B(z,p[2],q[2])),z)- 2*p[2]*diff(opeL(B(z,p[2],q[2])),p[2])-opeL(B(z,p[2],q[2]))):Checkingn:=8: normal(series(subs(B(z,p[2],q[2])=B_ser(n),H13B),z,n+1));
map(normal,subs(p[1]=0,q[1]=0,series (diff(diff(H_ser(n),p[1]),p[3]),z,n+1)));
factor(series(%%-%,z,n+1))An alternative expression of H13 (a bit heavier)H13Balt:=1/3/p[3]*opeL(z*diff(B(z,p[2],q[2]),z)-2*p[2]*diff(B(z,p[2],q[2]),p[2])) +2*z^2*q[2]/(1-z^2*p[2]*q[2])*diff(B(z,p[2],q[2]),p[2]):n:=8: normal(series(subs(B(z,p[2],q[2])=B_ser(n),H13Balt),z,n+1));
map(normal,subs(p[1]=0,q[1]=0,series (diff(diff(H_ser(n),p[1]),p[3]),z,n+1)));
factor(series(%%-%,z,n+1))H22B:=diff(B(z,p[2],q[2]),p[2]$2)The equation for BeqB:=-H13B+H22B+1/12*H1111B+1/2*H11B^2:Checkingn:=8: normal(series(subs(B(z,p[2],q[2])=B_ser(n), eqB),z,n+1))eqB:=convert(collect(convert(eqB,D),B,distributed,factor),diff):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 2" layout="Heading 2">What is determined by eqB? Apparently, everything apart from bipartite cubic maps (Remark 2.4). But we need to prescribe symmetry in p2 and q2</Text-field>unassign('coB')First, we do not prescribe symmetryB_form:=proc(N) local i,j,n:
add(add(add(coB[n,3*i-n,3*j-n]*p^(3*i-n)*q^(3*j-n)*z^n,i=ceil(n/3)..floor(n/2)),j=ceil(n/3)..floor(n/2)),n=2..N)+O(z^(N+1)):end:n:=4:map(factor,subs(p[3]=1,q[3]=1,p[2]=p,q[2]=q,B_ser(n)));B_form(n)Extracting the coeff of z^4n:=5: normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)){coeffs(coeff(%,z,n-1),[p,q])}solve(%,coB[4, 2, 2])assign(%)Now we run the next 4 commands repeatedlyn:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:={coeffs(coeff(ser,z,n),[p,q])}:solve(coe,{seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..floor(n/2)),j=ceil(n/3)..floor(n/2)) }):assign(%)n; indets(B_form(n-1))Many missing terms. Let us force symmetryunassign('coB')B_form:=proc(N) local i,j,n:
add(
add(coB[n,3*i-n,3*i-n]*p^(3*i-n)*q^(3*i-n)*z^n,i=ceil(n/3)..floor(n/2))
+add(add(coB[n,3*i-n,3*j-n]*(p^(3*i-n)*q^(3*j-n)+q^(3*i-n)*p^(3*j-n))*z^n,i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)),n=2..N)
+O(z^(N+1)):end:n:=7:B_form(n)n:=5: normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)){coeffs(coeff(%,z,n-1),[p,q])}solve(%,coB[4, 2, 2])assign(%)We run the next 6 lines repeatedly, until n=13, where no solution is found in the polynomials B[n,i,j]: the system is then more constrained, and we can determine a polynomial B[m,i,j] that was left undetermined, for some m<nn:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:=[coeffs(coeff(ser,z,n),[p,q])]:nops(%){seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) }; nops(%)solve(coe,{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) })assign(%)For n=13, there are more equations (namely 4) that unknowns B[n,i,j] (3). Hence we will determine some of the earlier unknown coefficientsn;solve(subsop(3=NULL,coe),{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) })assign(%)factor(coe)solve(%,{coB[5, 1, 1]})assign(%)Est-ce vrai ? Oui...coeff(coeff(coeff(convert(subs(p[3]=1,q[3]=1,p[2]=p,q[2]=q,B_ser(5)),polynom),z,5),p,1),q,1)coeff(coeff(coeff(convert(subs(p[3]=1,q[3]=1,p[2]=p,q[2]=q,B_ser(5)),polynom),z,2),p,1),q,1)factor(subs(coB[2,1,1]=%,(u^4 - 9*u^2*coB[2, 1, 1] + 5*u^2 + 54*coB[2, 1, 1]^2)/(5*u)))Then we go on, and meet a new additional equation for n= 16n:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:={coeffs(expand(coeff(convert(ser,polynom),z,n)),[p,q])}:nops(%){seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) }: nops(%)solve(coe,{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) })assign(%)n;solve({coe[7],seq(coe[i],i=1..5)},{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) })assign(%)factor(coe)solve(%,{coB[8, 1,1], coB[16, 2, 8]})assign(%)factor(coe)indets(B_form(n-1))n:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:={coeffs(expand(coeff(convert(ser,polynom),z,n)),[p,q])}:nops(%){seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) };nops(%)solve(coe,{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) })assign(%)n;solve({seq(coe[i],i=1..6)},{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) })assign(%)factor(coe);indets(%)solve({%%[2],%%[4]},{coB[19, 2, 2],coB[19, 2, 5]})assign(%)factor(coe); indets(%)solve(%%[2],{coB[11, 1, 1]})assign(%)indets(B_form(n-1))Let us do three more - no, two more only !n:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:={coeffs(expand(coeff(convert(ser,polynom),z,n)),[p,q])}:nops(%){seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) };nops(%)solve(coe,{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) });assign(%)Here we run into trouble already with n=21n;solve({seq(coe[i],i=1..8)},{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) })assign(%)factor(coe); indets(%)solve(op(2,%%),{coB[2, 1, 1]})assign(%)factor(coe); indets(%)solve(%%,{coB[21, 0, 3], coB[21, 0, 6], coB[21, 3, 3]})assign(%)factor(coe)indets(B_form(n-1))Let us go on...n:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:={coeffs(expand(coeff(convert(ser,polynom),z,n)),[p,q])}:nops(%){seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) };nops(%)solve(coe,{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) });n;solve({seq(coe[i],i=1..10)},{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) });assign(%)factor(coe); indets(%)solve(%%[2],{coB[14,1,1]})assign(%)factor(coe); indets(%)solve(%%,{coB[22, 2, 8], coB[22, 5, 5], coB[22, 5, 8]})assign(%)indets(B_form(n-1))n:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:={coeffs(expand(coeff(convert(ser,polynom),z,n)),[p,q])}:nops(%){seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) };nops(%)solve(coe,{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) });assign(%)n;solve({seq(coe[i],i=1..10)},{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) });assign(%)coe1:=factor(coe); indets(%);nops(%%)solve(coe1[2],{ coB[25, 2, 8]})assign(%)coe1:=factor(coe); indets(%);nops(%%)solve(coe1[2],{ coB[25, 5, 5]})assign(%)coe1:=factor(coe); indets(%);nops(%%)solve(coe1[2],{ coB[25, 2, 5]}); assign(%)coe1:=factor(coe); indets(%);nops(%%)solve(coe1,{coB[17, 1, 1],coB[25, 2, 2]}); assign(%)factor(coe)indets(B_form(n-1))n:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqB,diff))),z,n+1)):coe:={coeffs(expand(coeff(convert(ser,polynom),z,n)),[p,q])}:nops(%){seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) };nops(%)solve(coe,{seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) }):assign(%)n;indets(coe)solve(coe,{coB[20, 1, 1]} union {seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) }):assign(%)indets(B_form(n-1))Forcing symmetry and periodicity, we see that the only information that is missing is the GF of cubic bipartite mapsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 2" layout="Heading 2">A PDE in p2 and q2 only (Remark 2.5)</Text-field>indets(convert(eqB,D))The terms that contain a z-derivativecoeff(convert(eqB,D), D[1, 2](B)(z, p[2], q[2]));
coeff(convert(eqB,D), D[1, 3](B)(z, p[2], q[2]))The symmetric PDE: we first exchange p and q, and then use the symmetry of B. Let us test on one term of eqB that Maple is doing things as we wish.op(1,expand(eqB))subs({p[2]=q[2],q[2]=p[2],p[3]=q[3],q[3]=p[3]},%);
subs(B(z, q[2], p[2])=B(z, p[2], q[2]),%)eqB_sym:=convert(subs(B(z, q[2], p[2])=B(z, p[2], q[2]),subs({p[2]=q[2],q[2]=p[2],p[3]=q[3],q[3]=p[3]},eqB)),D):indets(%)We solve for the two terms with a z-derivativesolu:=op(solve({convert(eqB,D),eqB_sym},[D[1, 2](B)(z, p[2], q[2]),D[1, 3](B)(z, p[2], q[2])])):A new PDEres:=numer(factor(diff(op(2,solu[1]),q[2])-diff(op(2,solu[2]),p[2]))):nops(res)indets(res)eqBpq:=collect(res,B,factor):nops(%)Testn:=8: normal(series(eval(subs(B(z, p[2], q[2])=B_ser(n),convert(eqBpq,diff))),z,n+1))An antisymmetric equationres:=factor(subs(B(z, q[2], p[2])=B(z, p[2], q[2]),subs({p[2]=q[2],q[2]=p[2],p[3]=q[3],q[3]=p[3]},convert(eqBpq,diff)))/convert(eqBpq,diff)):indets(res)resHere we try to see what is determined by this new PDE. We prescribe symmetryunassign('coB')B_form:=proc(N) local i,j,n:
add(
add(coB[n,3*i-n,3*i-n]*p^(3*i-n)*q^(3*i-n)*z^n,i=ceil(n/3)..floor(n/2))
+add(add(coB[n,3*i-n,3*j-n]*(p^(3*i-n)*q^(3*j-n)+q^(3*i-n)*p^(3*j-n))*z^n,i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)),n=2..N)
+O(z^(N+1)):end:B_ser(2);B_form(2)n:=5:We run repeatedly the next 5 commandsn:=n+1: ser:=normal(series(eval(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1, B(z,p,q)=B_form(n),convert(eqBpq,diff))),z,n+1)):coe:={coeffs(coeff(ser,z,n),[p,q])}:vars:={seq(coB[n,3*i-n,3*i-n],i=ceil(n/3)..floor(n/2))} union {seq(seq(coB[n,3*i-n,3*j-n],i=ceil(n/3)..j-1),j=ceil(n/3)..floor(n/2)) }; nops(%)solve(coe,vars):assign(%)Many undetermined termsn;indets(B_form(n-1))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 series BB_ser(4)The PDE obtained for B is called eqBnops(eqB);op(11,eqB)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 1" layout="Heading 1">A PDE for the Ising GF of cubic maps (Section 3)</Text-field>The change of variable: first from I to B (Psi), then from B to I (Phi)den:=1-z^2*p*qFrom (z,p,q) to (t, nub, nuw)Psib:=[u^(1/3)*z/den,z*p,z*q]ConversePhib:=[t*s^(1/6)*(1-nub*nuw), nub/(t*s^(1/6)*(1-nub*nuw)), nuw/(t*s^(1/6)*(1-nub*nuw))]Checkingsimplify(subs(t=Psib[1], nub=Psib[2],nuw=Psib[3],u =s^(-1/2),Phib)) assuming (s>0,abs(p)<1/10,abs(q)<1/10,abs(z)<1/10)simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),Psib))The Jacobian matrix (Lemma 3.2)First rowfactor(diff(Psib[1],z)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));
factor(diff(Psib[1],p)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));
factor(diff(Psib[1],q)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));Second rowfactor(diff(Psib[2],z)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));
factor(diff(Psib[2],p)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));
factor(diff(Psib[2],q)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));Third rowfactor(diff(Psib[3],z)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));
factor(diff(Psib[3],p)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));
factor(diff(Psib[3],q)):simplify(subs(z=Phib[1], p=Phib[2],q=Phib[3],u =s^(-1/2),%));with(PDEtools):Using the dchange function does not change the name of the functionchange:=[z=Phib[1], p=Phib[2],q=Phib[3],u=1/sqrt(s)]The operator D_zcollect(convert(dchange(change,z*D[1](B)(z, p, q,u),newvars=[t,nub,nuw,s]),D),B,factor);The operator d/dpdchange(change,D[2](B)(z, p, q,u),newvars=[t,nub,nuw,s])
opdp:=proc(J)
-diff(J, nub)*t*s^(1/6)*(nub*nuw - 1) + diff(J, t)*s^(1/6)*t^2*nuw: end:The operator LopL:=proc(F)
2*z*diff(F, q)/(-p*q*z^2 + 1) + 2*z^2*q*diff(F, p)/(-p*q*z^2 + 1);
end proccollect(convert(dchange(change,opL(B(z,p,q,u)),newvars=[t,nub,nuw,s]),D),B,factor):
convert(%,diff)The image of L in the new variables is named LambdaopLambda:=proc(J)
-2*t^2*nuw*(nub*nuw - 1)*diff(J, nub) - 2*t^2*(nub*nuw - 1)*diff(J, nuw) + 2*t^3*(nuw^2 + nub)*diff(J, t): end:R:= factor(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1,eval(subs(B(z, p[2], q[2])=0,eqB))))factor(subs(change, u*z^2*q/(1-z^2*p*q)))factor(subs(change,R))factor(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1,eval(subs(B(z, p[2], q[2])=u^2/2*log(1/(1-z^2*p[2]*q[2])),eqB))))factor(subs(change,%))Qpap:=collect(%*s^(2/3)/t^5,[s,t],distributed,factor)Qpap := (32*nuw^8 + 10*nub*nuw^6 + 20*nub^2*nuw^4 + 32*nub^3*nuw^2 + 118*nuw^5 + 32*nub^4 + 108*nub*nuw^3 + 74*nub^2*nuw + 64*nuw^2 + 22*nub)*s*t^3 + (nuw^5 + 2*nuw^2 + nub)*s + 2*(2*nuw^4 + nub*nuw^2 + 2*nub^2 + 3*nuw)^2*t^3 + 2*nuw*(2*nuw^4 + nub*nuw^2 + 2*nub^2 + 3*nuw)The PDE for Ires:=factor(subs(p[2]=p,q[2]=q,p[3]=1,q[3]=1,eval(subs(B(z, p[2], q[2])=u^2*PsiI(z,p[2],q[2],u)+u^2/2*log(1/(1-z^2*p[2]*q[2])),eqB)))): indets(res)res1:=factor(s^(2/3)*dchange(change,res,newvars=[t,nub,nuw,s])):res2:=convert(res1,D):indets(res2)nops(res2)nops(op(2,res2))op(2,res2)eqI:=collect(subs(PsiI=II,res2),II,distributed,factor): indets(%); nops(%%)op(11,eqI)The version given in the paper (Theorem 3.3)opOmega:=proc(J) local res:
res:=nuw*t^2*diff(J,t)+t*(1-nub*nuw)*diff(J,nub):
-(nuw*t^2*diff(res,t)+t*(1-nub*nuw)*diff(res,nub))
+ 1/3*(t*diff(opLambda(J),t)+nuw*diff(opLambda(J),nuw)-nub*diff(opLambda(J),nub)-opLambda(J)):
end:The RHS of the equationRHS:=proc(J)
s/12*opLambda(opLambda(opLambda(opLambda(J)))) +1/2*(opLambda(opLambda(J)))^2+ t*(nuw+2*t^3*(2*nuw^4+nub*nuw^2+2*nub^2+3*nuw))*opLambda(opLambda(J))+t^5*Qpap: end:eqIpap:=-opOmega(II(nub, nuw, s, t))+RHS(II(nub, nuw, s, t)):res:=factor(convert(eqI,diff)/eqIpap)Note that the equation contains a factor t^2resa:=numer(factor(eqIpap)): nops(%)op(1,resa)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvThe PDE is called eqInops(eqI); op(21,eqI)<Text-field style="Heading 1" layout="Heading 1">Computing coefficients of the Ising series from the PDE? Let us first experiment.</Text-field>Giving almost no information: II is a series in t and that's it.
We will observe experimentally that if we bound the degree in nuw and nub, and impose one symmetry condition (and no constant term in t), then I seems to be characterized by the equation. This is then proved in Section 4.unassign('II1')IIform1:=proc(N) local n:
add(II1[n](nub,nuw,s)*t^n,n=0..N)+O(t^(N+1)): end:IIform1(1)n:=2: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1))ee:=collect(convert(op(3,op(1,%)),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[0](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree (by 0): II1_0 can be any constantk:=0: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}; indets(%)Fix the constant term to 0II1[0](nub, nuw, s):=0:Next termn:=3: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1)):ee:=collect(convert(coeff(%,t,n),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[1](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))indets(%)If we bound the total degree in nub and nuw by 1=n-2k:=n-2: eq:={coeffs(collect(eval(subs(II1[1](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])};solve(eq,indets(eq minus{s}))collect(subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k)),co)Impose symmetry -> 0II1[1](nub, nuw, s):=0:n:=4: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1)):ee:=collect(convert(coeff(%,t,n),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))indets(%)coeff(%%,co[1,1])If we bound the total degree in nub and nuw by 2k:=n-2: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])};solve(eq,indets(eq minus{s}))collect(subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k)),co)II1[2](nub, nuw, s):=0:n:=5: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1)):ee:=collect(convert(coeff(%,t,n),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[3](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))indets(%)coeff(%%,co[3,0])If we bound the total degree in nub and nuw by 3k:=n-2: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])};solve(eq,indets(eq) minus{s})res:=collect(subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k)),co)Forcing symmetry (when one of the nu's is 0)factor(subs(nub=0,nuw=nub,res)-subs(nuw=0,res))isolate(%,co[3, 0])collect(subs(%,res),[nub,nuw],factor)II1[3](nub, nuw, s):=%:n:=6: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1)):ee:=collect(convert(coeff(%,t,n),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))indets(%)If we bound the total degree in nub and nuw by 4=n-2k:=n-2: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s})res:=collect(subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k)),co)II1[n-2](nub, nuw, s):=%:n:=7: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1)):ee:=collect(convert(coeff(%,t,n),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))indets(%)coeff(%%,co[0,1])If we bound the total degree in nub and nuw by 4=n-2k:=n-2: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s})res:=collect(subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k)),co)Forcing symmetry (when one of the nu's is 0)factor(subs(nub=0,nuw=nub,res)-subs(nuw=0,res))isolate(%,co[n-2, 0])collect(subs(%,res),[nub,nuw],factor)II1[5](nub, nuw, s):=%:n:=8: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1)):ee:=collect(convert(coeff(%,t,n),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))indets(%)If we bound the total degree in nub and nuw by 6=n-2k:=n-2: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s})res:=collect(subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k)),co)Forcing symmetry (when one of the nu's is 0)factor(subs(nub=0,nuw=nub,res)-subs(nuw=0,res))II1[n-2](nub, nuw, s):=resn:=9: factor(series(eval(subs(II(nub,nuw,s,t)=IIform1(n),convert(eqI,diff))),t,n+1)):ee:=collect(convert(coeff(%,t,n),D),II1,factor)A polynomial solution ? Well, there seem to be somek:=8: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s}):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))indets(%)If we bound the total degree in nub and nuw by 7 =n-2k:=n-2: eq:={coeffs(collect(eval(subs(II1[n-2](nub, nuw, s)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k), convert(ee,diff))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq) minus{s})res:=collect(subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k)),co)Forcing symmetry (when one of the nu's is 0)factor(subs(nub=0,nuw=nub,res)-subs(nuw=0,res))II1[n-2](nub, nuw, s):=0LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 1" layout="Heading 1">Uniqueness and recursive calculation of I_n (Section 4)</Text-field>Property of the operator Omegan:='n':factor(opOmega(t^n*J(nub,nuw)))unassign('i','j'): factor(opOmega(cc[i,j]*t^n*nub^i*nuw^j)/(t^n)/t^2)image:=map(simplify,collect(%, [nuw,nub], distributed, factor),symbolic)Extracting the coeff of nub^k nuw^lunassign('k','l'): subs(i=k, j=l-2, op(1,image))+ subs(i=k+1,j=l-1,op(2,image))+subs(i=k-1,j=l,op(3,image)) +subs(i=k+2,j=l,op(4,image))+ subs(i=k,j=l+1,op(5,image))
rec:=map(Z->factor(Z/nub^k/nuw^l),%)The left-to-right recursionrec_left:=subs(k=i+1,l=j,rec)expr:=collect(solve(rec_left,cc[i,j]),[cc[i+1,j-2],cc[i + 2, j -1],cc[i+1, j + 1],cc[i+3, j]],factor)The top-down recursionrec_down:=subs(k=i,l=2+j,rec); subs(j=n,i=0,rec_down)op(opLambda)op(RHS)Effective procedureRHSn:=proc(n) option remember:
normal(coeff(series(RHS(IIser(n-3)),t,n+3),t,n+2)):
end:coI:=proc(n,i,j) option remember:
if n=0 then 0
elif n mod 3 <>0 then 0
elif (i-j mod 3) <> 0 then 0
elif i<0 or j<0 or i+j>n then 0
elif j=n then 3/n/(3+n) * coeff(RHSn(n),nuw,n+2)
elif i=n then coI(n,0,n)
elif i<j then coI(n,j,i)
else normal(3/2/(j-n)/(i-j-n)*coeff(coeff(RHSn(n),nub,i+1),nuw,j)
+ (i + 1 - n)*(i + 2*j - n)*coI(n,i + 1, j - 2)/(2*(j - n)*(i - j - n))
- (i + 2)*(2*i + 3 + j - 2*n)*coI(n,i + 2, j - 1)/((j - n)*(i - j - n))
+ (j + 1)*coI(n,i + 1, j + 1)/(j - n) + 3*(i + 3)*(i + 2)*coI(n,i + 3, j)/(2*(j - n)*(i - j - n))):
fi: end:coIn:=proc(n) local i,j,k: option remember:
if n=0 then 0
elif n mod 3 <>0 then 0
else add(add(coI(n,j+3*k,j)*nub^(j+3*k)*nuw^j,k=ceil(-2*j/3)..floor((n-2*j)/3)),j=0..n):
fi:
end:IIser:=proc(N) local n: option remember:
add(coIn(n)*t^n,n=1..N)+O(t^(N+1)): end:IIser(5)IIser(8);
subs(nub=0,nuw=0,%)Checking the equationIIser(30):n:=30:normal(series(eval(subs(II(nub, nuw, s, t)=IIser(n), eqIpap)),t,n+1))test: we restart from scratch the above procedures and see how much time it takes to compute 60 coeffs - well, rather 20, given the periodicityt1:=time(): IIser(60): time()-t1;test: we restart from scratch and see how much time it takes to compute for 72 edges)t1:=time(): IIser(72): time()-t1;t1:=time(): IIser(90): time()-t1;Maximal genus 30degree(IIser(90),s)120 edges, maximal genus 20t1:=time(): IIser(120): time()-t1;degree(IIser(120),s)<Text-field style="Heading 2" layout="Heading 2">The kernel of the n-th operator Omega_n, without the symmetry assumption: dimension 0 or 1, depending on the parity of n (Remark 4.4)</Text-field>opOmegan:=proc(n,J)
factor(opOmega(t^n*J)/t^(n+2)): end:Case n=0, polynomial solutions of degree at most k=8 in nuw and nubn:=0: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by n (hence 0 so far)k:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))n:=1: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq));subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by n: a non-trivial solutionk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq))subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))n:=2: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by n: no solutionk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))n:=3: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by n: one solutionk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))n:=4: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by n: no solutionk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))n:=5: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by n: a non-zero solutionk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))n:=6: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))n:=7: k:=8: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))factor(%)n:=8: k:=10: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=1..k),j=1..k))indets(%)Bounding the degree in each variable by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k),j=0..k))Bounding the total degree by nk:=n: eq:={coeffs(collect(eval(subs(J(nub, nuw)=add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k),opOmegan(n,J(nub,nuw)))),[nub,nuw],distributed,factor),[nub,nuw])}:solve(eq,indets(eq)):subs(%,add(add(nub^i*nuw^j*co[i,j],i=0..k-j),j=0..k))factor(%)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 have computed up to 120 edges (maximal genus 20)degree(IIser(120),s)degree(IIser(30),s)IIser(3)IIser(6)coeff(coeff(convert(IIser(12),polynom),t,12),s,2)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 1" layout="Heading 1">A PDE for the Ising series recording the face number rather than genus (if preferred)</Text-field>indets(eqI)Let It (nub, nuw, u, t) count Ising maps by edges (as before) and faces (u).res:=op(3,numer(factor(eval(subs(II(nub, nuw, s, t)=s*It(nub,nuw,1/sqrt(s),t*s^(1/6)), convert(eqI,diff)))))):nops(res); op(21,indets(res))resa:=simplify(subs(s=1/u^2,res),symbolic): op(21,indets(%))resb:=numer(factor(simplify(subs(t=t*u^(1/3),resa),symbolic))):indets(%)nops(resb)eqIt:=collect(resb,It,distributed,factor):nops(eqIt)op(11,eqIt)<Text-field style="Heading 1" layout="Heading 1">The monochromatic white case (Section 5.2)</Text-field>Mwser:=proc(n)
O(t^(n+1))+subs(t=0,nu=t,normal(subs(nuw=nu/t,convert(IIser(n),polynom)))): end:Mwser(10)Construction from the general equationopLamw:=proc(J) 2*t^2*t*diff(J,t): end:opOmw:=proc(J) 1/3*(2*t*diff(opLamw(J),t) - opLamw(J)) -t^2*diff (t^2*diff(J,t),t): end:eqM:=collect(op(2,numer(factor(convert(eval(-opOmw(M(t))+s/12* opLamw(opLamw(opLamw(opLamw(M(t))))) +1/2 *opLamw(opLamw(M(t)))^2 +t*(1+4*t^3)*opLamw(opLamw(M(t)))+t^5*(4+s) +8*t^8*(1+4*s)),D)))),M,distributed,factor)n:=26: normal(series(subs(M(t)=Mwser(n),convert(eqM,diff)),t,n))subs(M=N,collect(eqM,[(D@@4)(M)(t),(D@@3)(M)(t),(D@@2)(M)(t),D(M)(t)],factor))For the series with no periodicityeqMt:=collect(subs(t=t^(1/3),op(3,factor(eval(subs(M(t)=Mt(t^3),convert(eqM,diff)))))),Mt,factor)JSFHcollect(eqM,M,distributed,factor)The quadratic parteqMquad:=factor(216*D(M)(t)^2*t^7 + 144*D(M)(t)*(D@@2)(M)(t)*t^8+24*(D@@2)(M)(t)^2*t^9 );The constant termeqMcont:=(3*t^2*(32*s*t^3 + 8*t^3 + s + 4))The linear parteqMlin:=collect(eqM-%-%%,M,factor)factor((420*s*t^6 + 144*t^6 + 36*t^3 - 4)/4)A change of functioneqR:=collect(convert(numer(factor(eval(subs(M(t)=int((RR(t)+1/2/t-t^2)/2/t^3,t) ,convert(eqM,diff))))),D),RR,factor)The known Goulden and Jackson recursioncor:=proc(n,g) option remember: local i,h:
if g=0 and n=0 then 1
elif g=0 and n=-1 then -1/2
elif n<1 then 0
elif g<0 then 0
else 1/(n+1)*(4*n*(3*n-2)*(3*n-4)*cor(n-2,g-1) + 4 *add(add((3*i+2)*(3*(n-2-i)+2)*cor(i,h)*cor(n-2-i,g-h),h=0..g),i=-1..n-1)) fi: end:N:=3:add(add(cor(n,g)*t^(3*n)*s^g/6/n,g=0..3*n),n=1..N)Mwser(N*3)series(%%-%,t,10)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 2" layout="Heading 2">Double specialization: The planar white case (Remark 5.7)</Text-field>(subs(s=0,eqM))N:=62: subs(s=0,Mwser(N))ser:=series(2*t*diff(%,t),t,N+1)serT:=series(subs(t=T^(1/3),%),T,(N+1)/3)seriestoalgeq(serT,C(T)); map(factor,op(1,%));latex(%)seriestolist(serT)seq(2*8^n/(n+1)/(n+2) *binomial(3*n/2,n),n=1..10)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<Text-field style="Heading 2" layout="Heading 2">Double specialization: The unicellular white case (Remark 5.7)</Text-field>eqUw:=collect(op(2,numer(factor(convert(eval(-opOmw(Uw(t))+1/12* opLamw(opLamw(opLamw(opLamw(Uw(t)))))) +t^5,D)))),Uw,distributed,factor); latex(%)seq(coeff(coeff(coIn(3*(2*g-1)),s,g),nuw,3*(2*g-1)),g=1..5)From Walsh and Lehman I (but where precisely?) or Chapuy (New combinatorial identity..., Cor. 8)seq((6*g-4)!/12^g/g!/(3*g-2)!,g=1..5)Checking the DEN:=30: series(eval(subs(Uw(t)= add((6*g-4)!/12^g/g!/(3*g-2)!*t^(3*(2*g-1)),g=1..N), convert(eqUw,diff))),t,N+1)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<Text-field style="Heading 1" layout="Heading 1">Unicellular maps (Sections 5.3)</Text-field>One face: 3(2g-1) edges in genus g.
Let us look at the corresponding polynomials, and compare the values that we have obtained with known results from the literature.seq(coeff(coIn(3*(2*g-1)),s,g),g=1..3)One face, monochromatic whiteseq(coeff(coeff(coIn(3*(2*g-1)),s,g),nuw,3*(2*g-1)),g=1..5)From Walsh and Lehman I or Chapuy (New combinatorial identity..., Cor. 8)seq(2*(6*g-3)!/12^g/g!/(3*g-2)!/(12*g-6),g=1..5)seq(2*(6*g-3)!/12^g/g!/(3*g-2)!,g=1..5)One face, bipartite: nice numbers as wellseq(subs(nuw=0,nub=0,coeff(coIn(3*(2*g-1)),s,g)),g=1..6); ifactor([%])From Goupil and Schaefferseq(1/2/3^(g-1)*(2*g)!*(3*g-2)!/(g-1)!/g!^2/(6*g-3),g=1..5)seq(1/2/3^(g-1)*(2*g)!*(3*g-2)!/(g-1)!/g!^2,g=1..5)A PDE for Ising cubic maps with 1 face, counted by edges (t) and monochromatic white and black edges can be obtained from the PDE derived in the previous section for Ising maps counted by edges and faces.eqU:=collect(convert(subs(u=0,numer(factor(1/u*eval(subs(It(nub, nuw, u, t)=u*U(nub,nuw,t)+u^2*Reste(nub,nuw,u,t), convert(eqIt,diff)))))),D),U,distributed,factor): nops(res)But in fact this PDE can be written explicitely by specializing the one in the papereqUpap:=convert(-opOmega(U(nub, nuw, t))+ 1/12*opLambda(opLambda(opLambda(opLambda(U(nub,nuw,t))))) +t^5*(nuw^5+2*nuw^2+nub),D):factor(eqUpap/eqU)Generating coefficients from the PDERHSU:=proc(J)
1/12*opLambda(opLambda(opLambda(opLambda(J))))+t^5*(nuw^5+2*nuw^2+nub):
end:RHSUn:=proc(n) option remember:
normal(coeff(series(RHSU(User(n-3)),t,n+3),t,n+2)):
end:coU:=proc(n,i,j) option remember:
if n=0 then 0
elif n mod 3 <>0 then 0
elif n mod 2 =0 then 0
elif (i-j mod 3) <> 0 then 0
elif i<0 or j<0 or i+j>n then 0
elif j=n then 3/n/(3+n) * coeff(RHSUn(n),nuw,n+2)
elif i=n then coU(n,0,n)
elif i<j then coU(n,j,i)
else normal(3/2/(j-n)/(i-j-n)*coeff(coeff(RHSUn(n),nub,i+1),nuw,j)
+ (i + 1 - n)*(i + 2*j - n)*coU(n,i + 1, j - 2)/(2*(j - n)*(i - j - n))
- (i + 2)*(2*i + 3 + j - 2*n)*coU(n,i + 2, j - 1)/((j - n)*(i - j - n))
+ (j + 1)*coU(n,i + 1, j + 1)/(j - n) + 3*(i + 3)*(i + 2)*coU(n,i + 3, j)/(2*(j - n)*(i - j - n))):
fi: end:coUn:=proc(n) local i,j,k: option remember:
if n=0 then 0
elif n mod 3 <>0 then 0
else add(add(coU(n,j+3*k,j)*nub^(j+3*k)*nuw^j,k=ceil(-2*j/3)..floor((n-2*j)/3)),j=0..n):
fi:
end:User:=proc(N) local n: option remember:
add(coUn(n)*t^n,n=1..N)+O(t^(N+1)): end:User(16)series(%-add(t^(3*(2*g-1))*coeff(coIn(3*(2*g-1)),s,g),g=1..3),t,17)test: we restart from scratch the above procedures and see how much time it takes to compute 60 coeffs - well, rather 20, given the periodicityt1:=time(): User(60): time()-t1;test: we restart from scratch and see how much time it takes to compute 120 coeffs (only 20 of them are non-zero)t1:=time(): User(120): time()-t1;t1:=time(): User(200): time()-t1;User(201):201/3LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 2" layout="Heading 2">Second PDE: Using Chapuy's bijection with trisections</Text-field>Two new operatorsOp_w:=proc(Pol)
2*t^2*( (nub+nuw^2)*(t*diff(Pol,t)-nuw*diff(Pol,nuw)-nub*diff(Pol,nub))
+(nub+nuw/nub)*nub*diff(Pol,nub)
+(nuw^2+1/nuw)*nuw*diff(Pol,nuw)): end:Op_b:=proc(Pol)
2*t^2*( (nub^2+nuw)*(t*diff(Pol,t)-nuw*diff(Pol,nuw)-nub*diff(Pol,nub))
+(nuw+nub/nuw)*nuw*diff(Pol,nuw)
+(nub^2+1/nub)*nub*diff(Pol,nub)): end:U_ch:=proc(g) option remember:
if g=1 then t^3/6*(2+nuw^3+nub^3)
else
factor(1/(2*g)*1/6*(Op_w(Op_w(Op_w(U_ch(g-1)))) +Op_b(Op_b(Op_b(U_ch(g-1)))))):fi:
end:U_ch(1);expand(U_ch(2)/t^9);expand(U_ch(3)/t^15);Comparison4200*nub^9*nuw^3 + 4200*nub^3*nuw^9 + 22715*nub^10*nuw + 14175*nub^7*nuw^4 + 14175*nub^4*nuw^7 + 22715*nub*nuw^10 + 43750*nub^8*nuw^2 + 43750*nub^2*nuw^8 + 105*nub^10*nuw^4 + 105*nub^4*nuw^10 + 1155*nub^11*nuw^2 + 525*nub^8*nuw^5 + 525*nub^5*nuw^8 + 1155*nub^2*nuw^11 + 99680*nub^7*nuw + 99680*nub*nuw^7 + 51205*nub^9 + 39200*nuw^5*nub^5 + 87500*nub*nuw^4 + 137200*nuw^4*nub^4 + 151375*nuw^5*nub^2 + 151375*nub^5*nuw^2 + 52465*nub^6 + 51205*nuw^9 + 52465*nuw^6 + 87500*nub^4*nuw + 67900*nuw^2*nub^2 + 8400*nub*nuw + 39200/3*nuw^3 + 560/3 + 50050/3*nuw^12 + 5005/3*nuw^15 + 50050/3*nub^12 + 39200/3*nub^3 + 5005/3*nub^15 + 492800/3*nub^3*nuw^3 + 251125/3*nub^6*nuw^3 + 9800/3*nub^6*nuw^6 + 251125/3*nub^3*nuw^6factor(%%/%)unassign('i','j','m')The underlying recursion: applying the operator to a monomial t^(3*m-6)*nuw^j*nub^ires:=collect(normal((Op_w(Op_w(Op_w(t^(3*m-6)*nuw^j*nub^i))) +Op_b(Op_b(Op_b(t^(3*m-6)*nuw^j*nub^i))))/t^(3*m-6)/nuw^j/nub^i/t^6),[nub,nuw],distributed,simplify,factor)Let us invert this to obtain the coefficient of t^(3m)nub^k nuw^lldegree(res,nub); degree(res,nuw)collect(add(add(subs(i=k-a,j=l-b,coeff(coeff(res,nub,a),nuw,b)*U(3*m-6,i,j)),a=-3..6),b=-3..6),U,factor)coeff(%,U(3*m-6,k,l))We recycle this to get a procedureUco:=proc(n,l,k) local m: option remember:
if (n mod 3) <>0 then 0
elif (n mod 2) =0 then 0
elif n=3 and k=0 and l=0 then 1/3
elif n=3 and k=0 and l=3 then 1/6
elif n=3 and k=3 and l=0 then 1/6
elif n=3 then 0
else m:=n/3:
((-72*k^2*l + 72*k^2*m - 72*k*l^2 + 288*k*l*m + 72*l^2*m - 120*k^2 - 480*k*l + 216*k*m - 120*l^2 + 216*l*m - 408*k - 408*l + 144*m - 288)*Uco(3*m - 6, k, l) - 8*(k + 2 - 3*m)*(k - 3*m + 6)*(-3*m + 4 + k)*Uco(3*m - 6, k, l - 6) + (-8*k^3 + 72*k^2*l - 72*k^2*m - 288*k*l*m + 216*k*m^2 + 216*l*m^2 + 216*m^3 + 408*k*l - 288*k*m - 648*l*m - 864*m^2 + 80*k + 456*l + 912*m - 168)*Uco(3*m - 6, k, l - 3) + 8*(l + 3)*(l + 2)*(l + 1)*Uco(3*m - 6, k, l + 3) - 8*(-3*m + 2 + l)*(-3*m + 6 + l)*(-3*m + 4 + l)*Uco(3*m - 6, k - 6, l) + 24*(l + 1)*(l^2 - 6*l*m + 9*m^2 + 9*l - 27*m + 19)*Uco(3*m - 6, k - 5, l + 1) - 24*(l^2 - 6*l*m + 9*m^2 + 7*l - 21*m + 11)*(k + 2 - 3*m)*Uco(3*m - 6, k - 4, l - 1) - 24*(l + 2)*(l + 1)*(l + 5 - 3*m)*Uco(3*m - 6, k - 4, l + 2) + (72*k*l^2 - 288*k*l*m + 216*k*m^2 - 8*l^3 - 72*l^2*m + 216*l*m^2 + 216*m^3 + 408*k*l - 648*k*m - 288*l*m - 864*m^2 + 456*k + 80*l + 912*m - 168)*Uco(3*m - 6, k - 3, l) + 8*(l + 3)*(l + 2)*(l + 1)*Uco(3*m - 6, k - 3, l + 3) - 24*(l + 6 + k - 6*m)*(k*l - 3*k*m - 3*l*m + 9*m^2 + 3*k + 3*l - 18*m + 8)*Uco(3*m - 6, k - 2, l - 2) - 24*(l + 1)*(3*k*l - 6*k*m - l^2 + 3*l*m - 9*m^2 + 10*k - 7*l + 27*m - 17)*Uco(3*m - 6, k - 2, l + 1) - 24*(k^2 - 6*k*m + 9*m^2 + 7*k - 21*m + 11)*(-3*m + 2 + l)*Uco(3*m - 6, k - 1, l - 4) + (72*k^2*l - 144*k^2*m + 72*k*l^2 - 576*k*l*m + 648*k*m^2 - 144*l^2*m + 648*l*m^2 + 168*k^2 + 672*k*l - 1512*k*m + 168*l^2 - 1512*l*m + 960*k + 960*l - 576*m + 672)*Uco(3*m - 6, k - 1, l - 1) + 24*(l + 2)*(l + 1)*(-l - 5 + k + 3*m)*Uco(3*m - 6, k - 1, l + 2) + 24*(k + 1)*(k^2 - 6*k*m + 9*m^2 + 9*k - 27*m + 19)*Uco(3*m - 6, k + 1, l - 5) + 24*(k + 1)*(k^2 - 3*k*l - 3*k*m + 6*l*m + 9*m^2 + 7*k - 10*l - 27*m + 17)*Uco(3*m - 6, k + 1, l - 2) + 24*(k + 1)*(l + 1)*(k + 2 + l)*Uco(3*m - 6, k + 1, l + 1) + 8*(k + 3)*(k + 2)*(k + 1)*Uco(3*m - 6, k + 3, l) - 24*(k + 2)*(k + 1)*(k + 5 - l - 3*m)*Uco(3*m - 6, k + 2, l - 1) + 8*(k + 3)*(k + 2)*(k + 1)*Uco(3*m - 6, k + 3, l - 3) - 24*(k + 2)*(k + 1)*(k + 5 - 3*m)*Uco(3*m - 6, k + 2, l - 4))/12/(m+1)*2:
fi: end:
expand(U_ch(2)/t^9);Checkg:=1: n:=3*(2*g-1): add(add(Uco(n,i,j)*nub^i*nuw^j,i=0..n),j=0..n);
factor(%-U_ch(g)/t^n)A big oneg:=10: n:=3*(2*g-1): add(add(Uco(n,i,j)*nub^i*nuw^j,i=0..n),j=0..n):
factor(%-U_ch(g)/t^n)k:=8: User(3*k):add(U_ch(n),n=1..k):series(%%-%,t,3*k+1)i:=0: j:=0: ifactor([seq(Uco(3*(2*g-1),3*(2*g-1)-i,j),g=1..6)])unassign('k','l'):The complete recurrence relation (here, m must be of the form 2g-1)id:=-U(3*m,k,l)+((-72*k^2*l + 72*k^2*m - 72*k*l^2 + 288*k*l*m + 72*l^2*m - 120*k^2 - 480*k*l + 216*k*m - 120*l^2 + 216*l*m - 408*k - 408*l + 144*m - 288)*U(3*m - 6, k, l) - 8*(k + 2 - 3*m)*(k - 3*m + 6)*(-3*m + 4 + k)*U(3*m - 6, k, l - 6) + (-8*k^3 + 72*k^2*l - 72*k^2*m - 288*k*l*m + 216*k*m^2 + 216*l*m^2 + 216*m^3 + 408*k*l - 288*k*m - 648*l*m - 864*m^2 + 80*k + 456*l + 912*m - 168)*U(3*m - 6, k, l - 3) + 8*(l + 3)*(l + 2)*(l + 1)*U(3*m - 6, k, l + 3) - 8*(-3*m + 2 + l)*(-3*m + 6 + l)*(-3*m + 4 + l)*U(3*m - 6, k - 6, l) + 24*(l + 1)*(l^2 - 6*l*m + 9*m^2 + 9*l - 27*m + 19)*U(3*m - 6, k - 5, l + 1) - 24*(l^2 - 6*l*m + 9*m^2 + 7*l - 21*m + 11)*(k + 2 - 3*m)*U(3*m - 6, k - 4, l - 1) - 24*(l + 2)*(l + 1)*(l + 5 - 3*m)*U(3*m - 6, k - 4, l + 2) + (72*k*l^2 - 288*k*l*m + 216*k*m^2 - 8*l^3 - 72*l^2*m + 216*l*m^2 + 216*m^3 + 408*k*l - 648*k*m - 288*l*m - 864*m^2 + 456*k + 80*l + 912*m - 168)*U(3*m - 6, k - 3, l) + 8*(l + 3)*(l + 2)*(l + 1)*U(3*m - 6, k - 3, l + 3) - 24*(l + 6 + k - 6*m)*(k*l - 3*k*m - 3*l*m + 9*m^2 + 3*k + 3*l - 18*m + 8)*U(3*m - 6, k - 2, l - 2) - 24*(l + 1)*(3*k*l - 6*k*m - l^2 + 3*l*m - 9*m^2 + 10*k - 7*l + 27*m - 17)*U(3*m - 6, k - 2, l + 1) - 24*(k^2 - 6*k*m + 9*m^2 + 7*k - 21*m + 11)*(-3*m + 2 + l)*U(3*m - 6, k - 1, l - 4) + (72*k^2*l - 144*k^2*m + 72*k*l^2 - 576*k*l*m + 648*k*m^2 - 144*l^2*m + 648*l*m^2 + 168*k^2 + 672*k*l - 1512*k*m + 168*l^2 - 1512*l*m + 960*k + 960*l - 576*m + 672)*U(3*m - 6, k - 1, l - 1) + 24*(l + 2)*(l + 1)*(-l - 5 + k + 3*m)*U(3*m - 6, k - 1, l + 2) + 24*(k + 1)*(k^2 - 6*k*m + 9*m^2 + 9*k - 27*m + 19)*U(3*m - 6, k + 1, l - 5) + 24*(k + 1)*(k^2 - 3*k*l - 3*k*m + 6*l*m + 9*m^2 + 7*k - 10*l - 27*m + 17)*U(3*m - 6, k + 1, l - 2) + 24*(k + 1)*(l + 1)*(k + 2 + l)*U(3*m - 6, k + 1, l + 1) + 8*(k + 3)*(k + 2)*(k + 1)*U(3*m - 6, k + 3, l) - 24*(k + 2)*(k + 1)*(k + 5 - l - 3*m)*U(3*m - 6, k + 2, l - 1) + 8*(k + 3)*(k + 2)*(k + 1)*U(3*m - 6, k + 3, l - 3) - 24*(k + 2)*(k + 1)*(k + 5 - 3*m)*U(3*m - 6, k + 2, l - 4))/6/(m+1)unassign('i','j')recij:=collect(subs(k=3*m-i,l=j,id),U,factor)unassign('g'):subs(i=0,j=0,recij)Some terms must be 0: U(3m-6,3m-k,l) where l<0 or l-k>-6. And for l-k=-6, only the case l=0 is non zerosubs(seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..20),l=-20..-1),seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..l+5),l=0..20),
seq(U(3*m-6, 3*m-l-6,l)=0,l=1..20), %);
subs(m=2*g-1,%)subs(i=2,j=1,recij):subs(seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..20),l=-20..-1),seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..l+5),l=0..20),
seq(U(3*m-6, 3*m-l-6,l)=0,l=1..20),%):
subs(m=2*g-1,%)Nombres plus simples, donc ? Ben oui...seq(Uco(6*g - 3, 6*g - 5, 1),g=1..10)subs(i=3,j=0,recij):subs(seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..20),l=-20..-1),seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..l+5),l=0..20),seq(U(3*m-6, 3*m-l-6,l)=0,l=1..20),U(3*m-6, 3*m-8,1)=0,%):
map(factor,subs(m=2*g-1,%))subs(i=4,j=2,recij):subs(seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..20),l=-20..-1),seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..l+5),l=0..20),seq(U(3*m-6, 3*m-l-6,l)=0,l=1..20),U(3*m-6, 3*m-8,1)=0,%):
map(factor,subs(m=2*g-1,%))The first non-trivial case with j oddsubs(i=5,j=1,recij):subs(seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..20),l=-20..-1),seq(seq(U(3*m-6, 3*m-k,l)=0,k=-20..l+5),l=0..20),seq(U(3*m-6, 3*m-l-6,l)=0,l=1..20),U(3*m-6, 3*m-8,1)=0,%);
map(factor,subs(m=2*g-1,%))Let's experiment morei:=0: j:=0: liste:=[seq((Uco(3*(2*g-1),3*(2*g-1)-i,j)/(6*g)!*12^g*g!*(3*g)!),g=1..30) ]with(CurveFitting):factor(RationalInterpolation([seq(g,g=1..30)],liste,g))Now i+j=3 (d=1)i:=3: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-2)!*12^g*g!*(3*g-1)!,g=1..20) ]:factor(RationalInterpolation([seq(g,g=1..20)],liste,g))Now i+j=6 (d=2)i:=4: j:=2: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-6)!*12^g*g!*(3*g-3)!,g=1..30) ]:factor(RationalInterpolation([seq(g,g=1..30)],liste,g))i:=5: j:=1: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-6)!*12^g*g!*(3*g-3)!,g=1..30) ]:factor(RationalInterpolation([seq(g,g=1..30)],liste,g))i:=6: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-6)!*12^g*g!*(3*g-3)!,g=1..30) ]:factor(RationalInterpolation([seq(g,g=1..30)],liste,g))Now i+j=9 (d=3)i:=5: j:=4: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=6: j:=3: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=7: j:=2: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=8: j:=1: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=9: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))Now i+j=12 (d=4)i:=7: j:=5: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=8: j:=4: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=9: j:=3: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=10: j:=2: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=11: j:=1: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=12: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-8)!*12^g*g!*(3*g-4)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))Now i+j=15 (d=5)i:=9: j:=6: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-10)!*12^g*g!*(3*g-5)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=10: j:=5: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-10)!*12^g*g!*(3*g-5)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=11: j:=4: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-10)!*12^g*g!*(3*g-5)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=12: j:=3: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-10)!*12^g*g!*(3*g-5)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=13: j:=2: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-10)!*12^g*g!*(3*g-5)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=14: j:=1: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-10)!*12^g*g!*(3*g-5)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))i:=15: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-10)!*12^g*g!*(3*g-5)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g))Now i+j=18=3*6i:=10: j:=8: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=11: j:=7: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=12: j:=6: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=13: j:=5: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=14: j:=4: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=15: j:=3: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=16: j:=2: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=17: j:=1: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)i:=18: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-12)!*12^g*g!*(3*g-6)!,g=2..30) ]:factor(RationalInterpolation([seq(g,g=2..30)],liste,g));degree(%)Now i+j=21 (d=7) avec d\303\251nominateur devin\303\251i:=11: j:=10: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=12: j:=9: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=13: j:=8: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=14: j:=7: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=15: j:=6: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=16: j:=5: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(PolynomialInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=17: j:=4: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(PolynomialInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=18: j:=3: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..30) ]:factor(PolynomialInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=19: j:=2: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..40) ]:factor(RationalInterpolation([seq(g,g=3..40)],liste,g)); degree(%)i:=20: j:=1: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..40) ]:factor(RationalInterpolation([seq(g,g=3..40)],liste,g)); degree(%)i:=21: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-14)!*12^g*g!*(3*g-7)!,g=3..40) ]:factor(PolynomialInterpolation([seq(g,g=3..40)],liste,g)); degree(%)Now i+j=24 (d=8), avec d\303\251nominateur devin\303\251i:=13: j:=11: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=14: j:=10: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=15: j:=9: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=16: j:=8: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..30) ]:factor(PolynomialInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=17: j:=7: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..30) ]:factor(PolynomialInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=18: j:=6: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..30) ]:factor(PolynomialInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=19: j:=5: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..40) ]:factor(RationalInterpolation([seq(g,g=3..40)],liste,g)); degree(%)i:=20: j:=4: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..40) ]:factor(RationalInterpolation([seq(g,g=3..40)],liste,g)); degree(%)i:=21: j:=3: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..40) ]:factor(PolynomialInterpolation([seq(g,g=3..40)],liste,g)); degree(%)i:=22: j:=2: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..40) ]:factor(PolynomialInterpolation([seq(g,g=3..40)],liste,g)); degree(%)i:=23: j:=1: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..40) ]:factor(PolynomialInterpolation([seq(g,g=3..40)],liste,g)); degree(%)i:=24: j:=0: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/(6*g-15)!*12^g*g!*(3*g-8)!,g=3..40) ]:factor(PolynomialInterpolation([seq(g,g=3..40)],liste,g)); degree(%)Now i+j=21, avec fonction gammaGAMMA(10-3/2)i:=11: j:=10: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/GAMMA(3*g-j/2-3/2)/(16/3)^g*g!,g=1..10) ]factor(RationalInterpolation([seq(g,g=1..10)],liste,g)); degree(%)i:=12: j:=9: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/GAMMA(3*g-j/2-2)/(16/3)^g*g!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)i:=13: j:=8: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/GAMMA(3*g-j/2-3/2)/(16/3)^g*g!,g=1..20) ]:factor(RationalInterpolation([seq(g,g=1..20)],liste,g)); degree(%)i:=12: j:=9: liste:=[seq(Uco(3*(2*g-1), 3*(2*g-1)-i,j)/GAMMA(3*g-j/2-2)/(16/3)^g*g!,g=3..30) ]:factor(RationalInterpolation([seq(g,g=3..30)],liste,g)); degree(%)A few more attemptsunassign('g');D-finiteness in t ?N:=40: liste:=subs(nuw=2,nub=2,[0,seq(U_ch(g)/t^(3*(2*g-1)),g=1..N)]): indets(%)ser:=listtoseries(liste,s): nops(ser)map(factor,op(1,op(1,listtorec(liste,c(g)))))seriestodiffeq(ser,A(s))collect(convert(op(1,op(1,seriestodiffeq(ser,A(s)))),D),A,factor)N:=60: liste:=subs(nuw=3,nub=3,[0,seq(U_ch(g)/t^(3*(2*g-1)),g=1..N)]): indets(%)ser:=listtoseries(liste,s): nops(ser)map(factor,op(1,op(1,listtorec(liste,c(g)))))collect(op(1,op(1,seriestodiffeq(ser,A(s)))),A,factor)N:=60: liste:=subs(nuw=4,nub=4,[0,seq(U_ch(g)/t^(3*(2*g-1)),g=1..N)]):ser:=listtoseries(liste,s): nops(ser)map(factor,op(1,op(1,listtorec(liste,c(g)))))collect(op(1,op(1,seriestodiffeq(ser,A(s)))),A,factor)N:=70: liste:=subs(nuw=2,nub=1,[0,seq(U_ch(g)/t^(3*(2*g-1)),g=1..N)]): indets(%)ser:=listtoseries(liste,s): nops(ser)map(factor,op(1,op(1,listtorec(liste,c(g)))))LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 2" layout="Heading 2">Are numbers nice?</Text-field>for g from 1 to 3 do N:=3*(2*g-1): print(ifactor([coeffs(coeff(convert(User(N),polynom),t,N))])): od:for g from 1 to 4 do N:=3*(2*g-1): print(ifactor([coeffs(subs(nuw=nu,nub=nu,coeff(convert(User(N),polynom),t,N)))])): od:with(LRETools)?hypergeomsolsBipartite (no monochromatic edge)liste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,0) ,g=1..10)]op(1,listtorec(liste,c(h)));
map(factor,op(1,%))hypergeomsols(%,c(h),{c(0) = 1/3}); seq(2*(3*h-2)!*(2*h-3)!/3^h/(h)!/(h-2)!/(h-1)!/liste[h+1],h=2..7)Few monochromatic edges3 monochromatic black edgesliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,3) ,g=1..20)]:op(1,listtorec(liste,c(h)));
map(factor,op(1,%))hypergeomsols(%,c(h),{});seq(1/4*(3*h-1)!*(2*h-1)!*(3*h-2)/3^h/(h)!/(h-1)!/(h-1)!/liste[h+1],h=2..7)6 monochromatic black edgesliste:=[seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,6) ,g=1..20)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))hypergeomsols(%,c(h),{});9 monochromatic black edgesliste:=[seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,9) ,g=1..20)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))hypergeomsols(%,c(h),{});1 monochromatic black edge, 1 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,1),nub,1) ,g=1..20)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))hypergeomsols(%,c(h),{});seq(9/2*(3*h-2)!*(2*h-1)!/3^h/(h)!/(h-1)!/(h-2)!/liste[h+1],h=2..7)1 monochromatic black edges, 4 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,4),nub,1) ,g=1..20)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))hypergeomsols(%,c(h),{});seq(3/32*(3*h-1)!*(2*h-1)!*(18*h^3 - 21*h^2 - 21*h + 16)/3^h/(h+1)!/(h-1)!/(h-2)!/liste[h+1],h=2..7)2 monochromatic black edges, 2 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,2),nub,2) ,g=1..20)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))hypergeomsols(%,c(h),{});seq(3/16*(3*h-2)!*(2*h-1)!*(54*h^3 - 63*h^2 - 57*h + 56)/3^h/(h+1)!/(h-1)!/(h-2)!/liste[h+1],h=2..7)3 monochromatic black edges, 3 whiteliste:=[seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,3),nub,3) ,g=1..30)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))hypergeomsols(%,c(h),{});Now many monochromatic edgesOnly monochromatic black edges (3n)liste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,3*(2*g-1)) ,g=1..30)]:op(1,listtorec(liste,c(h)));
map(factor,op(1,%))hypergeomsols(%,c(h),{c(1) = 1/6});seq(144^h*GAMMA(h - 1/2)*GAMMA(h + 1/6)*GAMMA(h - 1/6)/(288*Pi^(3/2)*GAMMA(h + 1)),h=1..6);
seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,3*(2*g-1)) ,g=1..6)ifactor([seq((6*g-5)!/12^g/g!/(3*g-3)!/liste[g+1],g=2..6)])seq((6*g-4)!/12^g/(3*g-2)!/g!,g=1..6)liste:=[seq(GAMMA(3*g+1/2)*(16/3)^g/g!/((6*g-4)!/12^g/(3*g-2)!/g!)/sqrt(Pi),g=1..30)]factor(RationalInterpolation([seq(g,g=1..30)], liste,g));(3n-2) monochromatic black edges, 1 whiteliste:=[seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,1),nub,3*(2*g-1)-2) ,g=1..20)]3n-3 monochromatic black edges, 0 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,3*(2*g-1)-3) ,g=1..20)]:op(1,listtorec(liste,c(h)));
map(factor,op(1,%))hypergeomsols(%,c(h),{c(1) = 1/3});ifactor([seq(2/3*(6*g-3)!/12^g/g!/(3*g-2)!/liste[g+1],g=2..6)])ifactor([seq(1/3*(6*g-2)!/12^g/g!/(3*g-1)!/liste[g+1],g=2..6)])(3n-4) monochromatic black edges, 2 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,2),nub,3*(2*g-1)-4) ,g=1..20)]:op(1,listtorec(liste,c(h)));
map(factor,op(1,%))hypergeomsols(%,c(h),{c(2)=5});ifactor([seq(6*(6*g-6)!/12^g/(g-1)!/(3*g-3)!/liste[g+1],g=2..6)])(3n-5) monochromatic black edges, 1 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,1),nub,3*(2*g-1)-5) ,g=1..20)]:op(1,listtorec(liste,c(h)));
map(factor,op(1,%))factor(hypergeomsols(%,c(h),{}));ifactor([seq(6*(12*g^2 - 18*g + 5)*(6*g-6)!/12^g/(g)!/(3*g-3)!/liste[g+1],g=2..6)])(3n-5) monochromatic black edges, 4 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,4),nub,3*(2*g-1)-5) ,g=1..20)]:op(1,listtorec(liste,c(h)));
map(factor,op(1,%))factor(hypergeomsols(%,c(h),{}));ifactor([seq(12*(6*g-8)!/12^g/(g-1)!/(3*g-4)!/liste[g+1],g=2..6)])(3n-6) monochromatic black edges, 0 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,0),nub,3*(2*g-1)-6) ,g=1..30)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))factor(hypergeomsols(%,c(h),{}));ifactor([seq(2*(4*g-5)*(12*g^2-19*g+6)*(6*g-6)!/12^g/(g)!/(3*g-3)!/liste[g+1],g=2..6)])monochromatic black edges (3n-6), 3 whiteliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,3),nub,3*(2*g-1)-6) ,g=1..30)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))factor(hypergeomsols(%,c(h),{}));([seq(96*(6*g-8)!/12^g/(g-1)!/(3*g-5)!/liste[g+1],g=2..6)])A case with a single bicoloured edgeliste:=[0,seq(coeff(coeff(coeff(convert(User(3*(2*g-1)),polynom),t,3*(2*g-1)),nuw,10),nub,3*(2*g-1)-11) ,g=1..30)]:op(1,listtorec(liste,c(h))):
map(factor,op(1,%))factor(hypergeomsols(%,c(h),{}));([seq(15120*(6*g-14)!/12^g/(g-2)!/(3*g-7)!/liste[g+1],g=4..10)])10!/5!/2; factor(%/12^2)ifactor(105)Could the series be D-finite in t for all values of nuw and nub? nuw0:=1: nub0:=1: g:=20: ser:=series(subs(t=t^(1/6),expand(subs(nuw=nuw0,nub=nub0,1/t^3*convert(User(3*(2*g-1)),polynom)))),t,g):seriestolist(ser):op(1,listtorec(%,c(h)))rec:=map(factor,op(1,%))hypergeomsols(%,c(h),{c(0) = 2/3});For nuw=nub ?nuw0:=2: nub0:=nuw0: g:=30: ser:=series(subs(t=t^(1/6),expand(subs(nuw=nuw0,nub=nub0,1/t^3*convert(User(3*(2*g-1)),polynom)))),t,g):seriestolist(ser):op(1,listtorec(%,c(h)))rec:=map(factor,op(1,%))hypergeomsols(%,c(h),{c(0) = 3});Here the order increasesnuw0:=3: nub0:=nuw0: g:=30: ser:=series(subs(t=t^(1/6),expand(subs(nuw=nuw0,nub=nub0,1/t^3*convert(User(3*(2*g-1)),polynom)))),t,g):seriestolist(ser):op(1,listtorec(%,c(h))):rec:=map(factor,op(1,%))hypergeomsols(%,c(h),{});No guess herenuw0:=4: nub0:=nuw0: g:=40: ser:=series(subs(t=t^(1/6),expand(subs(nuw=nuw0,nub=nub0,1/t^3*convert(User(3*(2*g-1)),polynom)))),t,g):seriestolist(ser):op(1,listtorec(%,c(h))):rec:=map(factor,op(1,%))hypergeomsols(%,c(h),{});Hard particlesnuw0:=1: nub0:=0: g:=40: ser:=series(subs(t=t^(1/6),expand(subs(nuw=nuw0,nub=nub0,1/t^3*convert(User(3*(2*g-1)),polynom)))),t,g):seriestolist(ser):op(1,listtorec(%,c(h))):rec:=map(factor,op(1,%))hypergeomsols(%,c(h),{});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<Text-field style="Heading 1" layout="Heading 1">A PDE for rooted Ising maps<Font foreground="[255,0,255]"> </Font>(Section 7.1)</Text-field>opLambda:=proc(J)
-2*t^2*nuw*(nub*nuw - 1)*diff(J, nub) - 2*t^2*(nub*nuw - 1)*diff(J, nuw) + 2*t^3*(nuw^2 + nub)*diff(J, t): end:opOmega:=proc(J) local res:
res:=nuw*t^2*diff(J,t)+t*(1-nub*nuw)*diff(J,nub):
-(nuw*t^2*diff(res,t)+t*(1-nub*nuw)*diff(res,nub))
+ 1/3*(t*diff(opLambda(J),t)+nuw*diff(opLambda(J),nuw)-nub*diff(opLambda(J),nub)-opLambda(J)):
end:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvIf J is a function of t^3: we divide by t^2 and the replace t by t^{1/3}.This is the operator Lambda_tildecollect(subs(t=t^(1/3),expand(opLambda(Jt(t^3,nub,nuw))/t^2)),Jt,factor)opLambda_t:=proc(Jt)
6*t*(nuw^2 + nub)*diff(Jt, t) - 2*nuw*(nub*nuw - 1)*diff(Jt, nub) + (-2*nub*nuw + 2)*diff(Jt, nuw): end:Now with Omega (first part)res:=subs(t=t^(1/3),collect(1/3/t^2*(t*diff(opLambda(Jt(t^3,nub,nuw)),t)+nuw*diff(opLambda(Jt(t^3,nub,nuw)),nuw)-nub*diff(opLambda(Jt(t^3,nub,nuw)),nub)-opLambda(Jt(t^3,nub,nuw))),Jt,factor))The paper versionres_bis:=collect(convert(1/3*(3*t*diff(opLambad_t(Jt(t,nub,nuw)),t)+nuw*diff(opLambad_t(Jt(t,nub,nuw)),nuw)-nub*diff(opLambad_t(Jt(t,nub,nuw)),nub)+opLambad_t(Jt(t,nub,nuw))),D),Jt,factor)factor(res-res_bis)Still with Omega (second part)res:=nuw*t^2*diff(Jt(t^3,nub,nuw),t)+t*(1-nub*nuw)*diff(Jt(t^3,nub,nuw),nub)resd:=subs(t=t^(1/3),collect((nuw*t^2*diff(res,t)+t*(1-nub*nuw)*diff(res,nub))/t^2,Jt,factor))Comparing with the version in the paperres1:=3*nuw*t*diff(Jt(t,nub,nuw),t)+(1-nuw*nub)*diff(Jt(t,nub,nuw),nub)res2:=collect(convert(nuw*res1+ 3*nuw*t*diff(res1,t)+(1-nuw*nub)*diff(res1,nub),D),Jt,factor)normal(res2-resd)opOm_t:=proc(J) local res1:
res1:=3*nuw*t*diff(Jt,t)+(1-nuw*nub)*diff(Jt,nub):
1/3*(3*t*diff(opLambda_t(Jt(t,nub,nuw)),t)+nuw*diff(opLambda_t(Jt(t,nub,nuw)),nuw)-nub*diff(opLambda_t(Jt(t,nub,nuw)),nub)+opLambda_t(Jt(t,nub,nuw)))-nuw*res1+ 3*nuw*t*diff(res1,t)+(1-nuw*nub)*diff(res1,nub): end:Now for Lambda2_tilderes:=collect(subs(t=t^(1/3),expand(opLambda(opLambda(Jt(t^3,nub,nuw)))/t^4)),Jt,factor)The paper versionopLambda_tplus:=proc(c,Jt)
opLambda_t(Jt)+c*(nuw^2+nub)*Jt: end:opLambda2_t:=proc(Jt)
opLambda_tplus(4,opLambda_t(Jt)):
end:collect(convert(opLambda2_t(Jt(t,nub,nuw)),D),Jt,factor)-resNow for Lambda4_tilderes:=collect(subs(t=t^(1/3),expand(opLambda(opLambda(opLambda(opLambda(Jt(t^3,nub,nuw)))))/t^8)),Jt,factor):The paper versionopLambda4_t:=proc(Jt)
opLambda_tplus(12,opLambda_tplus(8,opLambda_tplus(4,opLambda_t(Jt)))):
end:collect(convert(opLambda4_t(Jt(t,nub,nuw)),D),Jt,factor)-resAnd QQpap := (32*nuw^8 + 10*nub*nuw^6 + 20*nub^2*nuw^4 + 32*nub^3*nuw^2 + 118*nuw^5 + 32*nub^4 + 108*nub*nuw^3 + 74*nub^2*nuw + 64*nuw^2 + 22*nub)*s*t^3 + (nuw^5 + 2*nuw^2 + nub)*s + 2*(2*nuw^4 + nub*nuw^2 + 2*nub^2 + 3*nuw)^2*t^3 + 2*nuw*(2*nuw^4 + nub*nuw^2 + 2*nub^2 + 3*nuw)subs(t=t^(1/3),t^5*Qpap/t^2)Q_t:=%/tLambda_tilde is almost commutingindets(convert(diff(opLambda_t(Jt(t,nub,nuw)),t),D))We now write the PDE on Itilde in a slightly formalized way before differentiating iteq_form:=-Omt(t)+e*t^2*Ht(t)+1/2*t^2*Jt(t)^2+(a*t+b)*t*Jt(t)+t*(c*t+d)We look at the third and fourth derivatives of this PDE, and then eliminate Jt=Lambda2_tildeens:=[op(subs(diff(Omt(t), t, t,t,t)=Om4, diff(Omt(t), t, t,t)=Om3, diff(Omt(t), t, t)=Om2, diff(Omt(t), t)=Om1, diff(Ht(t), t, t,t,t)=H4, diff(Ht(t), t, t,t)=H3, diff(Ht(t), t, t)=H2, diff(Ht(t), t)=H1, diff(Jt(t), t, t,t,t)=J4, diff(Jt(t), t, t,t)=J3, diff(Jt(t), t, t)=J2,diff(Jt(t), t)=J1,Jt(t)=Jt,{seq(diff(eq_form,t$i),i=3..4)}))]edg_nice:=factor(resultant(op(1,%),op(2,%),Jt))We substitute the expressions of the operators back into this equation without Lambda2_tildeedg1:=collect(convert(subs(Om4=diff(Omt(t), t, t,t,t), Om3=diff(Omt(t), t, t, t), Om2=diff(Omt(t), t, t), Om1=diff(Omt(t), t), J4=diff(Jt(t), t, t,t,t), J3=diff(Jt(t), t, t, t), J2=diff(Jt(t), t, t), J1=diff(Jt(t), t), H4=diff(Ht(t), t, t,t,t), H3=diff(Ht(t), t, t, t), H2=diff(Ht(t), t, t), H1=diff(Ht(t), t), edg_nice),D),[Jt],factor)edg2:=normal(eval(subs(Jt(t)=opLambda2_t(It(nub, nuw, t,s)),Omt(t)=opOm_t(It(nub, nuw, t,s)),Ht(t)=opLambda4_t(It(nub, nuw, t,s)),e=s/12,b=nuw,convert(edg1,diff)))):We have obtained a DE for the t-derivative of IteqItr:=collect(convert(factor(subs(It(nub, nuw, t,s)=int(Itr(nub, nuw, t,s),t),convert(edg2,diff))),D),Itr,factor):indets(eqItr)The Ising series for rooted cubic maps of arbitrary genus counted by edges/3, monochromatic white and black edges is denoted by IRres:=numer(factor(eval(subs(Itr(nub, nuw, t,s)=1/6/t*IR(nub, nuw, t,s),convert(eqItr,diff))))):eqIR:=collect(convert(res,D),IR,distributed,factor):The PDE on IR that we have obtained is of order 7indets(eqIR)Computing its degreeres:=numer(factor(eval(subs(IR(nub, nuw, t,s)=x*Itx(nub, nuw, t,s),convert(eqIR,diff))))):degree(res,x)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<Text-field style="Heading 1" layout="Heading 1">The planar case, first labelled and then rooted (Sections 5.1 and 7.2)</Text-field>Let us generate terms in the labelled planar series op(RHS)RHSP:=proc(J)
1/2*opLambda(opLambda(J))^2+t*(nuw+2*t^3*(2*nuw^4+nub*nuw^2+2*nub^2+3*nuw))*opLambda(opLambda(J))+t^5*subs(s=0,Qpap):
end:RHSPn:=proc(n) option remember:
normal(coeff(series(RHSP(Pser(n-3)),t,n+3),t,n+2)):
end:coP:=proc(n,i,j) option remember:
if n=0 then 0
elif n mod 3 <>0 then 0
elif (i-j mod 3) <> 0 then 0
elif i<0 or j<0 or i+j>n then 0
elif j=n then 3/n/(3+n) * coeff(RHSPn(n),nuw,n+2)
elif i=n then coP(n,0,n)
elif i<j then coP(n,j,i)
else normal(3/2/(j-n)/(i-j-n)*coeff(coeff(RHSPn(n),nub,i+1),nuw,j)
+ (i + 1 - n)*(i + 2*j - n)*coP(n,i + 1, j - 2)/(2*(j - n)*(i - j - n))
- (i + 2)*(2*i + 3 + j - 2*n)*coP(n,i + 2, j - 1)/((j - n)*(i - j - n))
+ (j + 1)*coP(n,i + 1, j + 1)/(j - n) + 3*(i + 3)*(i + 2)*coP(n,i + 3, j)/(2*(j - n)*(i - j - n))):
fi: end:coPn:=proc(n) local i,j,k: option remember:
if n=0 then 0
elif n mod 3 <>0 then 0
else add(add(coP(n,j+3*k,j)*nub^(j+3*k)*nuw^j,k=ceil(-2*j/3)..floor((n-2*j)/3)),j=0..n):
fi:
end:Pser:=proc(N) local n: option remember:
add(coPn(n)*t^n,n=1..N)+O(t^(N+1)): end:Pser(8)Comparing with the case s=0 of the global seriessubs(s=0,IIser(8))The PDE for the labelled series Pres:=factor(1/t^2*numer(factor(eval(subs(II(nub, nuw, s, t)=P(nub, nuw, t), s=0,convert(eqI,diff)))))): indets(%)nops(res)This is the equation for the periodic series in t^3eqP_per:=convert(res,D):In order to avoid high powers of t in the equation, we want to write a PDE for the aperiodic series P(t^(1/3))
We introduce operators Omegat for Omega/t^2 and similarly for LambdatopLambda(J(nub, nuw, t))/t^2opLambdat:=proc(J)
-2*nuw*(nub*nuw - 1)*diff(J, nub) - 2*(nub*nuw - 1)*diff(J, nuw) + 2*t*(nuw^2 + nub)*diff(J, t):end:op(opOmega)convert(collect(convert(factor(opOmega(J(nub, nuw, t))/t^2),D),J,factor),diff)opOmegat:=proc(J)
-(4*nuw*(nub*nuw - 1)*diff(J, nub))/3 + (-(2*nub*nuw)/3 + 2/3)*diff(J, nuw) + (2*t*(nuw^2 + nub)*diff(J, t))/3 - ((nub*nuw - 1)*(nub*nuw - 3)*diff(J, nub, nub))/3 + 2*(nub*nuw - 1)*(-nuw^2 + nub)*diff(J, nub, nuw)/3 - (2*t*(-nub*nuw^2 + nub^2 + 2*nuw)*diff(J, nub, t))/3 - (2*nuw*(nub*nuw - 1)*diff(J, nuw, nuw))/3 + (2*t*(nuw + 1)*(nuw^2 - nuw + 1)*diff(J, nuw, t))/3 + t^2*(-nuw^2 + 2*nub)*diff(J, t, t)/3: end:So we divide the equation by t^2, and then write P(t)=Pt(t^3).
LHSOmegat_It_expr:=collect(factor(subs(t=t^(1/3),eval(opOmegat(Pt(nub, nuw, t^3))))),Pt,factor)An interesting property: the t-derivative of this LHS only involves t-derivatives of Ptfactor(diff(%,t)): indets(%);An interesting seriesJ_expr:=factor(1/t^4*opLambda(opLambda(Pt(nub, nuw, t^3)))):This Jt will have the same property as PtJt_expr:=factor(subs(t=t^(1/3),J_expr))indets(Jt_expr)indets(factor(diff(Jt_expr,t)))The aperiodic equationeq_form:=-Omt_Pt(t)+1/2*t^2*Jt(t)^2+(a*t+b)*t*Jt(t)+t*(c*t+d)Strategy: write the first and second t-derivatives of this equation. Then Jt is the only term without t-derivatives. Eliminate it between both equationsens:=[op(subs(diff(Omt_Pt(t), t, t)=Om2, diff(Omt_Pt(t), t)=Om1, diff(Jt(t), t, t)=Jt2,diff(Jt(t), t)=Jt1,Jt(t)=Jt,{seq(diff(eq_form,t$i),i=1..2)}))]ed_nice:=factor(resultant(op(1,%),op(2,%),Jt))indets(ed_nice)We now get back to an expression in terms of Pted1:=collect(convert(subs(Om2=diff(Omt_Pt(t), t, t), Om1=diff(Omt_Pt(t), t), Jt2=diff(Jt(t), t, t), Jt1=diff(Jt(t), t), ed_nice),D),[Jt],factor)indets(%)ed2:=normal(eval(subs(Jt(t)=Jt_expr,Omt_Pt(t)=Omegat_It_expr,convert(ed1,diff)))):indets(ed2)We have obtained a DE for the t-derivative of Pted3:=collect(convert(factor(subs(Pt(nub, nuw, t)=int(Ptp(nub, nuw, t),t),convert(ed2,diff))),D),Ptp,factor):indets(ed3)aexpr:=2*(2*nuw^4 + nub*nuw^2 + 2*nub^2 + 3*nuw);
bexpr:=nuw;
cexpr:=2*(2*nuw^4 + nub*nuw^2 + 2*nub^2 + 3*nuw)^2;
dexpr:=2*nuw*(2*nuw^4 + nub*nuw^2 + 2*nub^2 + 3*nuw);
eqPtp:=subs(a=aexpr,b=bexpr,c=cexpr,d=dexpr,ed3):save(eqPtp,ED_Ptp):The Ising series for rooted cubic planar maps by counted edges/3, monochromatic white and black edges is denoted IRPres:=numer(factor(eval(subs(Ptp(nub, nuw, t)=1/6/t*IRP(nub, nuw, t),convert(eqPtp,diff))))):nops(res)eqIRP:=collect(convert(res,D),IRP,distributed,factor):nops(%)nops(collect(convert(res,D),IRP,factor))op(1,eqIRP)indets(eqIRP)Checking the PDE. The rooted seriesIRPser:=proc(n) collect(6*t*diff(subs(t=t^(1/3),convert(subs(s=0,Pser(n*3)),polynom)),t),t)+O(t^(n+1)): end:IRPser(3)n:=15:normal(series(subs(IRP(nub,nuw,t)= IRPser(n),convert(eqIRP,diff)),t,1+n))save(eqIRP,'eqIRP'):read('eqIRP'):indets(eqIRP)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 2" layout="Heading 2">Rational parametrization (Appendix)</Text-field>Derived from B. Eynard's book, Chapter 8First part: a, b, T in terms of the parameters x_param:=gamma*z+ a0+a1/z+a2/z^2;
y_param:=gamma/z+ b0+b1*z+b2*z^2collect(a*x_param-x_param^2+y_param,z,factor)eqs1:=[coeff(%,z,2),coeff(%,z,1),coeff(%,z,0),coeff(%,z,-1)-T/gamma]collect(b*y_param-y_param^2+x_param,z,factor)eqs2:=[coeff(%,z,-2),coeff(%,z,-1),coeff(%,z,0),coeff(%,z,1)-T/gamma]sol:=factor(solve({op(eqs1),op(eqs2)},{a,b,T,a1,b1,a2,b2})); nops(sol)map(factor,collect(numer(-op(2,op(1,sol))/gamma^2),gamma))Second part: Lagrange inversionexpr:=(gamma+a0 *u+ a1*u^2+a2*u^3)^4*diff( gamma*u+b2/u^2+b1/u,u)W0_param:=factor(subs(sol,coeff(expr,u,3)/4))map(factor,collect(-numer(%)/gamma^2,gamma))SymmetrizationW_param:=factor(W0_param+subs({a0=b0,b0=a0},W0_param))subs(gamma=sqrt(G),a0=Ab,b0=Aw,map(factor,collect(-numer(W_param)/gamma^2,gamma)))factor(subs(G=0,%))LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 2" layout="Heading 2">Checking the known parametric solution (Section 7.2)</Text-field>op(5,sol)nubp:= -op(2,subs(a0=Ab,b0=Aw,gamma=sqrt(G),%));
nuwp := subs({Aw=Ab,Ab=Aw},%)op(1,sol)tp:=subs(a0=Ab,b0=Aw,gamma=sqrt(G),-op(2,%)/(nubp*nuwp-1)^3);GF of cubic maps IRPp:=subs(a0=Ab,b0=Aw,gamma=sqrt(G),W_param/op(2,op(1,sol))^2)<Text-field style="Heading 3" layout="Heading 3">The minimal polynomial of IRPp?</Text-field>with(Groebner):Basis([subs(Aw=Ab,numer(IRPp-IS)), subs(Aw=Ab,numer(tp-t)), subs(Aw=Ab,numer(nuwp-3))], lexdeg([Ab,G],[Is,t])):B:=%:nops(B)map(indets,B)factor(op(1,B))Basis([numer(IRPp-IS), numer(tp-t), numer(nuwp-nuw), numer(nubp-nub)], lexdeg([Aw,Ab,G,collect(numer(nuwp-nuw),G); collect(numer(nubp-nub),G)factor(%%*(8*Aw + 4*nub)-%*(8*Ab + 4*nuw))collect(%,G)GsolA:=factor(solve(%,G))tA:=factor(subs(G=GsolA,tp)):nuwA:=factor(subs(G=GsolA,nuwp))res:=factor(resultant(numer(t-tA),numer(nuw-nuwA),Ab)): nops(%)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 3" layout="Heading 3">Some verification with known cases</Text-field>Let us check the bicoloured (planar) casefactor(subs(Aw=0,Ab=0,nubp));
factor(subs(Aw=0,Ab=0,nuwp));
tp0:=factor(subs(Aw=0,Ab=0,tp))IRPp0:=factor(subs(Aw=0,Ab=0,IRPp))algS:=factor(resultant(numer(t-tp0),numer(S-IRPp0),G))This should be planar bicoloured cubic maps with a weight t^n for 2n vertices and 3n edgesop(algeqtoseries(algS,t,S,10,true))series(%/2,t,11)CorrectMore generally, let us check the symmetric (planar) casenup:=factor(subs(Aw=A,Ab=A,nubp));
tps:=factor(subs(Aw=A,Ab=A,tp))IRPps:=factor(subs(Aw=A,Ab=A,IRPp))We first express G in terms of A and nuGsol:=solve(nup-nu,G)tpsA:=factor(subs(G=Gsol,tps));
IRPsA:=factor(subs(G=Gsol,IRPps));algSs:=collect(op(6,factor(resultant(numer(t-tpsA),numer(S-IRPsA),A))),S,factor):with(algcurves):genus(subs(nu=17,algSs),t,S)Correct degreedegree(algSs,S)map(factor,op(algeqtoseries(algSs,t,S,6,true)))This should be planar Ising cubic maps with a weight t^n for 2n vertices and 3n edges and nu for monochromatic edgesmap(factor,subs(nub=nu,nuw=nu,IRPser(6)))normal(series(%%-%,t,7))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 change of variables change:=[t=tp,nub=nubp,nuw=nuwp]with(PDEtools);res:=dchange(change,collect(eqIRP,IRP,factor),newvars=[Aw,Ab,G]):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Ynindets(res);nops(res)op(11,eqIRP);
denom(dchange(change,%,newvars=[Aw,Ab,G]));indets(IRPp)resa:=eval(subs(IRP(Ab, Aw, G)=IRPp,res)):We want to decide whether resa is zero.We first try the symmetric case...resb:=subs(Aw=A,Ab=A,resa):nresb:=numer(resb): Yip !indets(nresb)And then the complete checknresa:=numer(resa): %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<Text-field style="Heading 1" layout="Heading 1"></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 1" layout="Heading 1"></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 1" layout="Heading 1"></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv<Text-field style="Heading 1" layout="Heading 1"></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv