######################################################################
##InvMaj: Save this file as  InvMaj                                  #
## To use it, stay in the                                            #
##same directory, get into Maple (by typing: maple <Enter> )         #
##and then type:  read InvMaj<Enter>                                 #
##Then follow the instructions given there                           #
##                                                                   #
##Written by Andrew Baxter and Doron Zeilberger, Rutgers University  #
#[baxter,zeilberg] at math dot rutgers dot edu                       #
######################################################################
 
#Created: March 30, 2010
 
print(`Created:  March 30, 2010`):
print(`This version: Jan. 10,2011, using comments of referee Emilie Hogan`):
print(`[the corrections to the program do not effect the paper].`):
print(` This is InvMaj `):
print(`A Maple package that accompanies the article `):
print(`The Number of Inversions and the Major Index of Permutations `):
print(`are Asymptotically Joint-Independently-Normal`):
print(`by Andrew Baxter and Doron Zeilberger`):

print(`and also available from Baxter's Zeilberger's websites.`):
print(``):
print(`Please report bugs to zeilberg at math dot rutgers dot edu`):
print(``):
 print(`The most current version of this  package and paper`):
 print(` are  available from`):
 print(`http://www.math.rutgers.edu/~zeilberg/  .`):
 print(`For a list of the procedures type ezra();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):

with(combinat):

ezra1:=proc()

if args=NULL then
 print(` The supporting procedures are: `):
 print(`F `):

else
ezra(args):
fi:

end:

ezra:=proc()

if args=NULL then
 print(`The main procedures are: Check1, Check2, FM8,FM8m, frs, MOP, G `):
 print(` `):


elif nops([args])=1 and op(1,[args])=Check1 then
print(`Check1(L,n,i): verifies empirically (and hence rigorously), that the conjectured`):
print(`expressions for the leading terms of the factorial moments about the mean satisfy the recurrences`):
print(`that they are supposed to, regarding the first equation on p. 8 of the article`):
print(`do: Check1(FM8m(n,i),n,i) `):

elif nops([args])=1 and op(1,[args])=Check2 then
print(`Check2(L,n,i): verifies empirically (and hence rigorously), that the conjectured`):
print(`expressions for the leading terms of the factorial moments about the mean satisfy the recurrences`):
print(`that they are supposed to, regarding the second equation on p. 8 of the article`):
print(`do: Check2(FM8m(n,i),n,i) `):


elif nops([args])=1 and op(1,[args])=F then
print(`F(n,i,p,q) the weight-enumerator`):
print(`according to the weight p^inv(pi)q^(maj(pi))`):
print(`for the pair (inv,maj) defined over n-permutations`):
print(`that end in i. For example, try:`):
print(`F(4,2,p,q);`):

elif nops([args])=1 and op(1,[args])=FM8 then
print(`FM8(n,i): pre-computed table of the factorial moments`):
print(`about the means, as polynomial expressions in (n,i)`):
print(`for the pair (inv,maj) defined over n-permutations`):
print(`that end in i. The output is a list-of-lists, let's call`):
print(`it L, such that L[r][s] gives the (r,s)-mixed-factorial-moment`):
print(`about the mean for 1<=r,s<=8`):
print(`Try:`):
print(`FM8(n,i); `):

elif nops([args])=1 and op(1,[args])=FM8m then
print(`FM8m(n,i): pre-computed table of the leading terms of the`):
print(`factorial moments`):
print(`about the means, as polynomial expressions in (n,i)`):
print(`for the pair (inv,maj) defined over n-permutations`):
print(`that end in i. The output is a list-of-lists, let's call`):
print(`it L, such that L[r][s] gives the (r,s)-mixed-factorial-moment`):
print(`about the mean for 1<=r,s<=8`):
print(`Try:`):
print(`FM8m(n,i); `):

elif nops([args])=1 and op(1,[args])=frs then
print(`frs(r0,s0,n,i): The (r0,s0)-factorial moment of (inv,maj) over`):
print(`n-permutations that end in i.`):
print(`For example, try:`):
print(`frs(2,2,n,i);`):

elif nops([args])=1 and op(1,[args])=G then
print(`G(n,i,p,q) the prob. generating function`):
print(`with variables p (for inv) and q(for maj)`):
print(`for the pair (inv,maj) defined over n-permutations`):
print(`that end in i. For example, try:`):
print(`G(4,2,p,q);`):

elif nops([args])=1 and op(1,[args])=MOP then
print(`MOP(ope,p,q,N,I1,r,s,R,S,m0): inputs an operator`):
print(`ope(n,i,p,q,N^(-1),I1^(-1)) such that a sequence of discrete`):
print(`prob. gen. functions satisfies the recurrence`):
print(`p_{n,i}(p,q)=ope(n,i,p,q,N^(-1),I1^(-1))p_{n,i}(p,q),`):
print(` expressed in terms of the discrete variables n and i and`):
print(` (negative) shift-operator`):
print(`N^(-1) amd I1^(-1), and given symbols r and s, and respective`):
print(` shift-operators  R and S`):
print(`and a pos. integer m0, outputs an operator, let's call it`):
print(`Ope(r,s,n,i,N^(-1),I1^(-1),R^(-1),S^(-1)) `):
print(`such that the factorial moments f_{r,s}(n,i)`):
print(`satisfy the recurrence, up to order m0 in R^(-1) and S^(-1)`):
print(`f_{r,s}(n)=Ope(r,s,N^(-1),I1^(-1),R^(-1),S^(-1))f_{r,s}(n,i) .`):
print(`For example, try: `):
print(`MOP(I1/q+(q^(n-1)-1)/(n-1)*p^(n/2-i)/q^(n/2)/N,p,q,N,I1,r,s,R,S,3);`):
print(`MOP((p*q)^(n/2-i)/N/(n-1),p,q,N,I1,r,s,R,S,1);`):


else
print(`There is no ezra for`,args):
fi:
 
end:





ezExtra:=proc(): print(` F(n,i,p,q)`): 
print(`AllTay(R0,N0), DataSets(R0,N0), GP2a(S,i,n,d), GP2(S,i,n)`):
print(`GuessFrs(R0,n,i),  Nor(L,i,n) `):
print(`MOP(I1/q+(q^(n-1)-1)/(n-1)*p^(n/2-i)/q^(n/2)/N,p,q,N,I1,r,s,R,S,3);`):

print(`qbin(q,n,k), A(n,p,q), , B(n,p,q), , Bd(n,p,q) `):
print(`   `):
print(`FindLeading(r,s,Aee,Aeo,Aoe,Aoo), Manig(P,x,y), FM8(n,i), FM8m() `):
print(`FM8ee(), FM8eo(), FMoe(), FMoo(),  Bdokee(), Bdokeo()`):
print(`Check1(L,n,i), Check2(L,n,i)`):
end:


####Data

#FM8(n,i): pre-computed table of the factorial moments
#about the means, as polynomial expressions in (n,i)
#for the pair (inv,maj) defined over n-permutations
#that end in i. The output is a list-of-lists, let's call
#it L, such that L[r][s] gives the (r,s)-mixed-factorial-moment
#about the mean for 1<=r,s<=10
FM8:=proc(n,i):
[[1/4-1/2*i+1/2*i^2+1/8*n-1/2*n*i+1/8*n^2, -1/4+1/2*i-1/2*i^2-1/8*n+1/2*n*i-1/8
*n^2, 7/16-7/8*i+7/8*i^2+1/6*n-37/48*n*i-5/48*n*i^2+31/192*n^2+1/6*n^2*i-1/16*n
^2*i^2-1/48*n^3+1/48*n^3*i+1/24*n^3*i^2-1/192*n^4-1/24*n^4*i+1/96*n^5, -9/8+9/4
*i-9/4*i^2-1/4*n+13/8*n*i+5/8*n*i^2-7/32*n^2-n^2*i+3/8*n^2*i^2+1/8*n^3-1/8*n^3*
i-1/4*n^3*i^2+1/32*n^4+1/4*n^4*i-1/16*n^5, -763/96*i+419/1920*n+763/192-749/
4608*n^4-283/80*n*i^2-46943/69120*n^3+97403/17280*n^2*i-7255/3456*n^2*i^2+2209/
3456*n^3*i+841/576*n^3*i^2-5027/3456*n^4*i+24391/69120*n^5+1831/34560*n^2+763/
96*i^2-31/1440*n^5*i^2+17/1080*n^6*i+5/864*n^6*i^2-5/864*n^7*i-19/3456*n^4*i^2+
467/17280*n^5*i-2117/480*n*i+5/3456*n^8-17/4320*n^7-89/23040*n^6, 1175/32*i+221
/128*n+1345/1536*n^4+349/16*n*i^2+18143/4608*n^3-39803/1152*n^2*i+14675/1152*n^
2*i^2-3845/1152*n^3*i-1805/192*n^3*i^2+10735/1152*n^4*i-9991/4608*n^5+6089/2304
*n^2-1175/32*i^2-1175/64+31/96*n^5*i^2-17/72*n^6*i-25/288*n^6*i^2+25/288*n^7*i+
95/1152*n^4*i^2-467/1152*n^5*i+477/32*n*i-25/1152*n^8+17/288*n^7+89/1536*n^6, -\
81469/384*i-168781/7560*n+24467/17418240*n^9+35/124416*n^11-1981/1244160*n^10-\
4889/1451520*n^8*i-25256003/4976640*n^4-36464719/241920*n*i^2-110008091/4354560
*n^3+38249741/161280*n^2*i-1194851/13824*n^2*i^2+22863641/1244160*n^3*i+
84672949/1244160*n^3*i^2-2608649/38880*n^4*i+37097981/2488320*n^5-13479517/
483840*n^2+81469/384*i^2-1546801/414720*n^5*i^2+140987/51840*n^6*i+3103/3072*n^
6*i^2-2963869/2903040*n^7*i-44303/46080*n^4*i^2+243191/51840*n^5*i+81469/768+
1981/311040*n^9*i-14860751/241920*n*i+15767/1451520*n^7*i^2+2920357/11612160*n^
8-1958051/2903040*n^7-553859/829440*n^6+35/31104*n^9*i^2-259/34560*n^8*i^2-35/
31104*n^10*i, 141463/96*i+936181/4320*n-24467/622080*n^9-245/31104*n^11+13867/
311040*n^10+4889/51840*n^8*i+39464621/1244160*n^4+10110559/8640*n*i^2+14047273/
77760*n^3-10532401/5760*n^2*i+2275217/3456*n^2*i^2-32772887/311040*n^3*i-\
171996643/311040*n^3*i^2+21081347/38880*n^4*i-141463/192-71528807/622080*n^5+
4377487/17280*n^2-141463/96*i^2+4265527/103680*n^5*i^2-387149/12960*n^6*i-25963
/2304*n^6*i^2+1199869/103680*n^7*i+13769/1280*n^4*i^2-336301/6480*n^5*i-13867/
77760*n^9*i+2621111/8640*n*i-15767/51840*n^7*i^2-1156357/414720*n^8+758531/
103680*n^7+1522073/207360*n^6-245/7776*n^9*i^2+1813/8640*n^8*i^2+245/7776*n^10*
i], [-1/4+1/3*i-1/3*i^3+1/24*n-1/6*n*i+1/2*n*i^2+1/24*n^2-1/6*n^2*i, -1/3*i+1/
72*n+17/144+151/5184*n^4+11/12*n*i^2+83/864*n^3-2/3*i^3-1/2*n^2*i+3/4*n^2*i^2-1
/4*n^3*i-1/432*n^5+403/5184*n^2+1/2*i^2+1/2*i^4-1/4*n*i-n*i^3+1/1296*n^6, 11/12
*i-3/8*n-1/36*n^3*i^3+1/24*n^2*i^3-77/864*n^4-35/8*n*i^2-31/96*n^3+37/12*i^3+
293/144*n^2*i-113/48*n^2*i^2+5/6*n^3*i-1/16*n^3*i^2+1/144*n^4*i+1/96*n^5-515/
864*n^2-5/2*i^2+1/24*n^4*i^2-1/72*n^5*i-3/2*i^4+20/9*n*i+221/72*n*i^3-1/432*n^6
-1/24, -11/2*i+210437/86400*n+1/15552*n^9+1/4*n^3*i^3-5/24*n*i^4-1/8*n^2*i^4+1/
12*n^3*i^4-1/6*n^4*i^3+247/320+5/12*n^2*i^3+26087/115200*n^4+49/3*n*i^2+2805919
/3110400*n^3-12*i^3-965/144*n^2*i+1013/144*n^2*i^2-359/144*n^3*i-5/24*n^3*i^2+
23/144*n^4*i-64699/1036800*n^5+104993/34560*n^2+51/4*i^2+1/8*n^5*i^2-1/24*n^6*i
-29/144*n^4*i^2+5/144*n^5*i+19/4*i^4-97/8*n*i-19/2*n*i^3-29/86400*n^8+2771/
518400*n^7+383/23040*n^6, 1681/48*i-46013/3456*n-5/7776*n^9-467/288*n^3*i^3-5/
2592*n^8*i+25/12*n*i^4+5/4*n^2*i^4-5/6*n^3*i^4+8659/5184*n^4*i^3-28745/5184*n^2
*i^3-535/64-86441/207360*n^4-17671/288*n*i^2-1246187/622080*n^3+2359/48*i^3+
537019/25920*n^2*i-15497/720*n^2*i^2+367463/51840*n^3*i+13385/3456*n^3*i^2-\
16309/10368*n^4*i+31/2160*n^5*i^3-5/1296*n^6*i^3+1457/4608*n^5-167251/11520*n^2
-200/3*i^2-4339/3456*n^5*i^2+22267/51840*n^6*i-31/1440*n^6*i^2+17/3240*n^7*i+
401/576*n^4*i^2+4121/51840*n^5*i-35/2*i^4+265967/4320*n*i+5/864*n^7*i^2+35347/
1080*n*i^3+199/51840*n^8-2839/51840*n^7-25937/207360*n^6, -3485/16*i+177199679/
2540160*n+419639/91445760*n^9-17/233280*n^11+84919/68584320*n^10+5659/576*n^3*i
^3+329/8640*n^8*i-1349/80*n*i^4-11575/1152*n^2*i^4+1321/192*n^3*i^4-2977/216*n^
4*i^3+5/559872*n^12+213817/4320*n^2*i^3-5/144*n^7*i^3+5/288*n^6*i^4-19/1152*n^4
*i^4-31/480*n^5*i^4-1589639549/2194698240*n^4+76339/320*n*i^2-1077421/18289152*
n^3-1759/8*i^3-90433/1728*n^2*i+352979/5760*n^2*i^2-531451/34560*n^3*i-1208797/
34560*n^3*i^2+10169/960*n^4*i-17/320*n^5*i^3+329/2160*n^6*i^3+1043479/16128-\
4666099/3732480*n^5+2096051611/30481920*n^2+11563/32*i^2+355673/34560*n^5*i^2-\
683/192*n^6*i+509/3840*n^6*i^2+41/34560*n^7*i-2549/2304*n^4*i^2-6655/3456*n^5*i
-5/576*n^9*i+2443/32*i^4-180259/576*n*i-581/4320*n^7*i^2-91123/720*n*i^3-\
32584687/731566080*n^8+9050077/20321280*n^7+4048625/4478976*n^6+5/192*n^8*i^2,
783793/576*i-43800433/120960*n-507467/17418240*n^9+3031/1866240*n^11-231073/
8709120*n^10-112773469/1866240*n^3*i^3-4444453/8709120*n^8*i+10829/80*n*i^4+
30625/384*n^2*i^4-3647/64*n^3*i^4+23611189/207360*n^4*i^3-43441/96-35/186624*n^
12-534373/1280*n^2*i^3+1571833/2177280*n^7*i^3-35/96*n^6*i^4+133/384*n^4*i^4+
217/160*n^5*i^4+80009011/4354560*n^4-5472863/5760*n*i^2+22720753/435456*n^3+
629927/576*i^3-382367/181440*n^2*i-21612329/241920*n^2*i^2-72342749/3265920*n^3
*i+6472613/23040*n^3*i^2-231908233/3732480*n^4*i-696503/622080*n^5*i^3-539699/
207360*n^6*i^3+13554971/7464960*n^5-9796439/30240*n^2-65877/32*i^2+259/51840*n^
8*i^3-35/46656*n^9*i^3+35/31104*n^10*i^2-35/93312*n^11*i-11573737/138240*n^5*i^
2+3575317/124416*n^6*i+202499/414720*n^6*i^2-274459/362880*n^7*i-16463471/
1244160*n^4*i^2+7526897/373248*n^5*i+2376533/13063680*n^9*i-12663/32*i^4+
73913479/45360*n*i+266867/138240*n^7*i^2+198151783/362880*n*i^3+4292213/8709120
*n^8-7588267/2177280*n^7-2429903/373248*n^6-259/34560*n^9*i^2-778033/1451520*n^
8*i^2+1981/933120*n^10*i, -1264543/144*i+23208801713/12441600*n+87191774443/
564350976000*n^9-311833007/11287019520*n^11+25661587237/62705664000*n^10+
90707723/233280*n^3*i^3+8931983/1451520*n^8*i-13842323/12096*n*i^4-2297351/3456
*n^2*i^4+155570629/311040*n^3*i^4-154835377/155520*n^4*i^3+35/5184*n^11*i^2-35/
15552*n^12*i+15767/362880*n^7*i^4-259/8640*n^8*i^4+35/7776*n^9*i^4-35/3888*n^10
*i^3+39257599/23514624000*n^12+21888025/6048*n^2*i^3-11846069/1088640*n^7*i^3+
4223/768*n^6*i^4-60263/11520*n^4*i^4-421853/20736*n^5*i^4-6066074896633/
31352832000*n^4+14197601/3780*n*i^2-950276533163/1567641600*n^3-222317/36*i^3+
73333291/40320*n^2*i-186757801/181440*n^2*i^2+5031355513/6531840*n^3*i-\
153769831/68040*n^3*i^2+469379567/1451520*n^4*i+4564313/155520*n^5*i^3+123347/
3456*n^6*i^3+22989303067531/564350976000*n^5+107260302413/72576000*n^2+1202449/
96*i^2-30271/181440*n^8*i^3+8393/116640*n^9*i^3-29729/466560*n^10*i^2+3031/
155520*n^11*i+3085079/4320*n^5*i^2-7445489/31104*n^6*i-1888987/77760*n^6*i^2+
3443/243*n^7*i+445123439/1866240*n^4*i^2-38379049/207360*n^5*i-7242163/2612736*
n^9*i+233425/96*i^4-227622397/25920*n*i-5806337/241920*n^7*i^2-1379/67184640*n^
14+35/20155392*n^15+1614307/4408992000*n^13-118454351/45360*n*i^3-15321521071/
2939328000*n^8+195228575711/7054387200*n^7+9058058120143/188116992000*n^6+
423653/2177280*n^9*i^2+34997729/4354560*n^8*i^2-32623/725760*n^10*i+430695181/
138240], [7/16-5/8*i-1/8*i^2+1/2*i^3+1/4*i^4-1/12*n+23/48*n*i-29/48*n*i^2-1/2*n
*i^3-17/192*n^2+1/6*n^2*i+3/16*n^2*i^2-1/48*n^3+1/48*n^3*i+1/24*n^3*i^2-1/192*n
^4-1/24*n^4*i+1/96*n^5, 9/8*i-7/12*n-i^5+5/2*n*i^4-9/4*n^2*i^3+37/864*n^4-49/48
*n*i^2+67/288*n^3+3/2*i^3-7/24*n^2*i+33/16*n^2*i^2-31/48*n^3*i+5/6*n^3*i^2-1/12
*n^4*i-1/288*n^5-341/864*n^2-19/8*i^2+3/4*i^4+91/48*n*i-9/4*n*i^3-1/432*n^6-1/
24, -191/32*i+257/128*n-85/48*n^3*i^3+3/4*i^5-167/96*n*i^4+89/32*n^2*i^4+1/48*n
^3*i^4-1/24*n^4*i^3+19/24*n^2*i^3+47/4608*n^4-221/48*n*i^2-5501/4608*n^3-9/8*i^
3+6461/1152*n^2*i-7397/1152*n^2*i^2+1319/1152*n^3*i+31/64*n^3*i^2-457/1152*n^4*
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260518080548888971/27993600*n^3*i^6+7158613/116640*n^11*i^8-12894290547781/
52254720*n^8*i^8+19355/279936*n^12*i^8-878395/6*n*i^14-19355/69984*n^13*i^7-\
1225/60466176*n^21*i+715645/1679616*n^14*i^6+18767/60466176*n^20*i-16807/
45349632*n^19*i^2-27881/11337408*n^18*i^3+1741661831/18*i^11-\
161555520860000907734955271/7900913664000000*n^4+1382898581341/3265920*n^7*i^9-\
72178034297/155520*n^6*i^10+2663802538680785743/839808000*n^5*i^5-\
5646968986233707983/43545600*n*i^2+18361111745848483650137/365783040000*n^3+
138272746106429/864*i^3-122796403781717267233/979776000*n^2*i+
678007838873035527859/1828915200*n^2*i^2-3363332658264130504723/54867456000*n^3
*i-8946730061949693566089/54867456000*n^3*i^2+7164999124801017478519/
123451776000*n^4*i-1487174070847/777600*n^5*i^10+151600463/108*n^4*i^11+1225/
30233088*n^18*i^4-1225/15116544*n^19*i^3+32462071985/7776*n^2*i^12+5934172391/
405*n*i^12-336595/11664*n^10*i^9+1546659347/503884800*n^15*i^4+7615105/
362797056*n^18*i-574301839/6046617600*n^17*i^2-16548037/755827200*n^16*i^3+
337511388272257883/7776000*n^2*i^5-8898015052505987117/2015539200*n^5*i^3+37093
/5038848*n^17*i^4+92361632100615225209/63489484800*n^6*i^3+215985*i^14+
3322309541366761913279/1975228416000*n^5+515420897006363/4478976*n^4*i^8+
25303567/7836416409600*n^22-2107/26121388032*n^23+1225/313456656384*n^24+206045
/1944*n^6*i^12-206045/324*n^7*i^11+314146441/194400*n^8*i^10+
6643672581436051561/132710400-6125/839808*n^16*i^5+315/2*i^16+80872569282504421
/671846400*n^6*i^6-1490042841048598327/2015539200*n^5*i^6-18218628259734419/
3386105856000*n^12*i^2-126134926391417/141087744000*n^13*i+30853451676547/
21332466892800000*n^19+810014473925147/1919922020352000000*n^20-1110925691/
65303470080000*n^21-10960824887208347/134369280*n^7*i^5-102845947975/54*i^9-\
25936906068478471168757/2633637888000*n^2-7400751969130301/69120*i^2-\
52337344838075918051/2962842624000*n^8*i^3-3474200224595775277/395045683200*n^9
*i^3+814325746136188499/1580182732800*n^10*i^2+166903212435599389/3950456832000
*n^11*i+4727011676843/17010*n*i^9-234473110357997/17280*i^6+809750733634520513/
3359232000*n^6*i^5-210749441/24*i^12+3997558230111346198379/282175488000*n^5*i^
2-473453043151176133441/1269789696000*n^6*i-222503609779456220633/338610585600*
n^6*i^2+32946213739562138339/376233984000*n^7*i-655908546993305875931/
65840947200*n^4*i^2-21586903252120125619289/1975228416000*n^5*i-6175960/3*i^13-\
275623564995431143/118513704960*n^9*i-415350509/432*n^3*i^12+902003872945385/
13824*i^4+465496674317077451/2073600*n*i-2648928366787471740077/10158317568000*
n^7*i^2+131114559015247/90720*n^2*i^9-5813738439031/466560*n^5*i^9-788064868889
/2821754880*n^13*i^4+1892923256011/14814213120*n^14*i^3-567826696309/1679616000
*n^12*i^5-1047182402771/671846400*n^11*i^5-4069478423233/223948800*n^10*i^6+
1191086164069/13063680*n^9*i^7-2017799586203/806400*n^7*i^8+11379690149903/
6531840*n^8*i^7+383851175393/155520*n^6*i^9-9213046887881/349920*n^3*i^9+
19378584239/540*n*i^11+175514406020281581371/3071875232563200000*n^14-\
3108752987996352569/511979205427200000*n^15-75056619568902557/6826389405696000*
n^13-752371132661856061/2177280*n*i^3-76361454089787121033743113/
8776786378752000000*n^8-53220073157969152201363/2370274099200000*n^7+
11010019500857192866285789/28443289190400000*n^6+219167508038920378183/
23702740992000*n^9*i^2-285800652143280791017/7900913664000*n^8*i^2-\
2367983084003208341/2962842624000*n^10*i-495635/1679616*n^15*i^5+304045/18*n^3*
i^13-781579/167961600*n^17*i^3+78280037/1007769600*n^16*i^4-2126405213/3499200*
n^11*i^7-773606869/349920*n^9*i^9+5061645474408433/43200*n*i^5+2039726087/6480*
n^5*i^11-18757449043/544320*n^8*i^9+2924075/6*n^2*i^13]]:
end:
####End Data

with(combinat):


#The sum of p^inv(pi)*q^maj(pi) over all n-perms that end in i
F:=proc(n,i,p,q) local i1: option remember:

if n<1 then
 RETURN(FAIL):
fi:
if n=1 then
 if i=1 then
   RETURN(1):
 else
   RETURN(0):
 fi:
fi:

if i=n then
 RETURN(add(F(n-1,i1,p,q),i1=1..n-1)):
fi:

expand(p*F(n,i+1,p,q)+
p^(n-i)*(q^(n-1)-1)*F(n-1,i,p,q)):


end:



Gd:=proc(n,i,p,q):
expand(simplify(expand(F(n,i,p,q)/(n-1)!/(p*q)^(n-i+binomial(n-1,2)/2)))):
end:


inv:=proc(pi) local i,j,co:
co:=0:
for i from 1 to nops(pi) do
 for j from i+1 to nops(pi) do
  if pi[i]>pi[j] then
    co:=co+1:
  fi:
 od:
od:
co:
end:


maj:=proc(pi) local i,co:
co:=0:
for i from 1 to nops(pi)-1 do
  if pi[i]>pi[i+1] then
    co:=co+i:
  fi:
 od:

co:
end:



#CheckF(n): checks the recurrence scheme for F directly
CheckF:=proc(n) local p,q,gu,i,pi,T:
gu:=permute(n):

for i from 1 to n do
 T[i]:=0:
od:


for pi in  gu do
T[pi[n]]:=T[pi[n]]+p^(inv(pi))*q^(maj(pi)):
od:

evalb([seq(T[i]-F(n,i,p,q),i=1..n)]=[0$n]):

end:




#G(n,i,p,q) the prob. generating function
#with variables p (for inv) and q(for maj)
#for the pair (inv,maj) defined over n-permutations
#that end in i. For example, try:
#G(4,2,p,q);
G:=proc(n,i,p,q) local i1: option remember:

if n<1 then
 RETURN(FAIL):
fi:
if n=1 then
 if i=1 then
   RETURN(1):
 else
   RETURN(0):
 fi:
fi:

if i=n then
 RETURN(expand(add(simplify((p*q)^(n/2-i1)*G(n-1,i1,p,q)/(n-1)),i1=1..n-1))):
fi:

expand(simplify(G(n,i+1,p,q)/q+
p^(n-i)*(q^(n-1)-1)/(p*q)^(n/2)/(n-1)*G(n-1,i,p,q))):


end:


CheckG:=proc(n) local p,q,i:
[seq(simplify(Gd(n,i,p,q)-G(n,i,p,q)),i=1..n)]:
end:





#GuessPOL1V1(L,x,d): Given a list L, tries to find
#a polynomial P(x) of degree d such that L[i]=P(i) for i=1..nops(L)
#For example, try:
#GuessPOL1V1([seq(i^2,i=1..10)],x,2);
GuessPOL1V1:=proc(L,x,d) local eq,var,a,P,i:

#if nops(L)-d<3 then
if nops(L)-d<1 then
 print(`Insufficient data`):
 RETURN(FAIL):
fi:

P:=add(a[i]*x^i,i=0..d):
var:={seq(a[i],i=0..d)}:
eq:={seq(expand(subs(x=i,P)-L[i]),i=1..nops(L))}:
var:=solve(eq,var):

if var=NULL then
  RETURN(FAIL):
fi:

subs(var,P):

end:



#GP(L,x): Given a list L, tries to find
#a polynomial P(x) such that L[i]=P(i) for i=1..nops(L) .
#For example, try:
#GP([seq(i^2,i=1..10)],x);
GP:=proc(L,x) local d,gu,i:

#for d from 0 to nops(L)-3 do
for d from 0 to nops(L)-1 do
 gu:=GuessPOL1V1(L,x,d):
if gu<>FAIL then
 if {seq(expand(subs(x=i,gu)-L[i]),i=1..nops(L))}={0} then
   RETURN(gu):
 fi:
fi:
od:

FAIL:

end:



#Taylor11(f,p,q,r0,s0): inputs an expression f in p and q
#f(p,q)
#and outputs the list-of-lists L such that L[r][s]
#is the coefficient of x^r*y^s/r!/s! in f(1+x,1+y)
#for r=1..r0, s=1..s0 .
#For example, try: 
#Taylor11(1/p/q,p,q,3,3);
Taylor11:=proc(f,p,q,r0,s0) local x,y,F,gu,i,j,F1:
option remember:
F:=subs({p=1+x,q=1+y},f):
gu:=[]:
F:=taylor(F,x=0,r0+1):

for i from 1 to r0 do
 F1:=coeff(F,x,i):
 F1:=taylor(F1,y=0,s0+1):
 gu:=[op(gu),[seq(i!*j!*coeff(F1,y,j),j=1..s0)]]:
od:
gu:
end:



#frs(r0,s0,n,i): The (r0,s0)-factorial moment of (inv,maj) over
#n-permutations that end in i
#For example, try:
#frs(2,2,n,i);
frs:=proc(r0,s0,n,i) local gu,n1,i1,mu,p,q,mu1,pol,lu:

gu:=[]:
for n1 from r0+s0+4 to 2*r0+2*s0+10 do

mu:=[seq(Taylor11(G(n1,i1,p,q),p,q,r0,s0)[r0][s0],i1=1..n1)]:

mu1:=GP(mu,i):
if mu1=FAIL then
 RETURN(FAIL,mu):
fi:
gu:=[op(gu),mu1]:

od:

pol:=GP(gu,n):

if pol=FAIL then
print(gu):
 RETURN(FAIL):
fi:

pol:=expand(subs(n=n-r0-s0-3,pol)):


lu:=
{seq(seq(
evalb(subs({n=n1,i=i1},pol)=Taylor11(G(n1,i1,p,q),p,q,r0,s0)[r0][s0]),
i1=1..n1),n1=1..r0+s0+8)}:

if lu<>{true} then
print(`Something went wrong`):
print(pol,lu):
RETURN(FAIL):
fi:

pol:

end:


#frsT(n,i,R0): a table of f_{r,s}(n,i) for 1<=r,s<=R0
frsT:=proc(n,i,R0) local r0,s0:
[seq([seq(frs(r0,s0,n,i),s0=1..R0)],r0=1..R0)]:
end:


AllTay:=proc(R0,N0) local i1,n1,p,q:
option remember:
seq([seq(Taylor11(G(n1,i1,p,q),p,q,R0,R0),i1=1..n1)],n1=1..N0):
end:

#DataSets(R0,N0): all the data sets for f[r][s](n,i)
#for 1<=r,s<=r0 using (n,i) data with 1<=i<=n<=N0
DataSets:=proc(R0,N0) local gu,r0,s0,mu,n1,i1,gu1,gu11:
mu:=AllTay(R0,N0):

gu:=[]:

for r0 from 1 to R0 do
 gu1:=[]:
 for s0 from 1 to R0 do
  gu11:={seq(seq([i1,n1,mu[n1][i1][r0][s0]],i1=1..n1),n1=1..N0)}:
  gu1:=[op(gu1),gu11]:
od:
 gu:=[op(gu),gu1]:
od:

gu:

end:



#GP2a(S,i,n,d): guesses a polynomial in (n,i) of degree <=d
#that agrees with the data-set S
GP2a:=proc(S,i,n,d) local a, eq,var,pol,i1,j1:

pol:=add(add(a[i1,j1]*n^i1*i^j1,j1=0..d-i1),i1=0..d):
var:={seq(seq(a[i1,j1],j1=0..d-i1),i1=0..d)}:

if nops(S)-nops(var)<4 then
 RETURN(FAIL):
fi:

eq:={seq(subs({i=S[j1][1],n=S[j1][2]},pol)-S[j1][3],j1=1..nops(S))}:
var:=solve(eq,var):
if var=NULL then
  FAIL:
else
  expand(subs(var,pol)):
fi:
end:

#GP2(S,i,n): guesses a polynomial in (n,i)
#that agrees with the data-set S
GP2:=proc(S,i,n) local gu,d:

for d from 0 while nops(S)-binomial(d+1,2)>=10 do
#for d from 0 while nops(S)-binomial(d+1,2)>=1 do

gu:=GP2a(S,i,n,d):

if gu<>FAIL then
  RETURN(gu):
fi:

od:

FAIL:

end:



#GP2aT(S,i,n,d): guesses a polynomial in (n,i) of degree <=d
#that agrees with the data-set S
GP2aT:=proc(S,i,n,d) local a, eq,var,pol,i1,j1:

pol:=add(add(a[i1,j1]*n^i1*i^j1,j1=0..d-i1),i1=0..d):
var:={seq(seq(a[i1,j1],j1=0..d-i1),i1=0..d)}:

if nops(S)-nops(var)<1 then
 RETURN(FAIL):
fi:

eq:={seq(subs({i=S[j1][1],n=S[j1][2]},pol)-S[j1][3],j1=1..nops(S))}:
var:=solve(eq,var):
if var=NULL then
  FAIL:
else
  expand(subs(var,pol)):
fi:
end:

#GP2T(S,i,n): guesses a polynomial in (n,i)
#that agrees with the data-set S
GP2T:=proc(S,i,n) local gu,d:

for d from 0 while nops(S)-binomial(d+1,2)>=10 do

gu:=GP2aT(S,i,n,d):

if gu<>FAIL then
  RETURN(gu):
fi:

od:

FAIL:

end:



#GuessFrs(R0,n,i): guesses f_{r,s}(n,i) for r,s<=R0
GuessFrs:=proc(R0,n,i) local gu,N0,r0,s0:
N0:=3*R0+2:
gu:=DataSets(R0,N0):

[seq([seq(GP2(gu[r0][s0],i,n),s0=1..R0)],r0=1..R0)]:
end:




#Nor(L,i,n): normalizes a list of lists of moments
#in variables (i,n)
#assumed to be asymptotically normal
Nor:=proc(L,i,n) local lu,d1,r0,s0,T:
d1:=degree(L[2][2],{i,n}):
lu:=sqrt(coeff(L[2][2],n,d1)):
if lu=0 then
 RETURN(FAIL):
fi:

for r0 from 1 to nops(L)/2 do
for s0 from 1 to nops(L[r0])/2 do

T[2*r0,2*s0]:=L[2*r0][2*s0]/lu^(r0+s0)/
((2*r0)!/r0!/2^r0)/((2*s0)!/s0!/2^s0):

T[2*r0-1,2*s0-1]:=L[2*r0-1][2*s0-1]/lu^(r0+s0)/
((2*r0)!/r0!/2^r0)/((2*s0)!/s0!/2^s0):

T[2*r0-1,2*s0]:=L[2*r0-1][2*s0]/lu^(r0+s0)/
((2*r0)!/r0!/2^r0)/((2*s0)!/s0!/2^s0):

T[2*r0,2*s0-1]:=L[2*r0][2*s0-1]/lu^(r0+s0)/
((2*r0)!/r0!/2^r0)/((2*s0)!/s0!/2^s0):

od:
od:

[seq([seq(T[r0,s0],s0=1..nops(L))],r0=1..nops(L))]:

end:

#MOP(ope,p,q,N,I1,r,s,R,S,m0): inputs an operator
#ope(n,i,p,q,N^(-1),I1^(-1)) such that a sequence of discrete
#prob. gen. functions satisfies the recurrence
#p_{n,i}(p,q)=ope(n,i,p,q,N^(-1),I1^(-1))p_{n,i}(p,q), expressed in terms
#of the discrete variables n and i and (negative) shift-operator
#N^(-1) amd I1^(-1), and given symbols r and s, and respective shift-operators 
# R and S
#and a pos. integer m0, outputs an operator, let's call it
#Ope(r,s,n,i,N^(-1),I1^(-1),R^(-1),S^(-1)) such that the factorial moments
#f_{r,s}(n,i)
#satisfy the recurrence, up to order m0 in R^(-1) and S^(-1)
#f_{r,s}(n)=Ope(r,s,N^(-1),I1^(-1),R^(-1),S^(-1))f_{r,s}(n,i)
#For example, try:
#MOP(I1/q+(q^(n-1)-1)/(n-1)*p^(n/2-i)/q^(n/2)/N,n,i,p,q,N,I1,r,s,R,S,3);
MOP:=proc(ope2,p,q,N,I1,r,s,R,S,m0)   local z,w,gu,ope,a,b,j1,j2,
gu1,gu11,s1,mu:

ope:=subs({p=1+z,q=1+w},ope2):



for a from ldegree(ope,N) to degree(ope,N) do
for b from ldegree(ope,I1) to degree(ope,I1) do

gu:=coeff(coeff(ope,N,a),I1,b):


gu1:=taylor(gu,z=0,m0+5):

for j1 from 0 to m0 do
 gu11:=coeff(gu1,z,j1):
 gu11:=taylor(gu11,w=0,m0+5):

 for j2 from 0 to m0 do
   s1[a,b,j1,j2]:=coeff(gu11,w,j2):
 od:
od:
od:
od:


gu:=add(
add(
add(
add( s1[a,b,j1,j2]*
expand(r!/(r-j1)!*s!/(s-j2)!)
*R^(-j1)*S^(-j2)*N^(a)*I1^b,j1=0..m0),
 j2=0..m0),
 a=ldegree(ope2,N)..degree(ope2,N)),
 b=ldegree(ope2,I1)..degree(ope2,I1)):

gu:=expand(gu):
mu:=0:

for j1 from ldegree(gu,R) to degree(gu,R) do
for j2 from ldegree(gu,S) to degree(gu,S) do
for a from ldegree(gu,N) to degree(gu,N) do
for b from ldegree(gu,I1) to degree(gu,I1) do
 mu:=mu+
factor(coeff(coeff(coeff(coeff(gu,R,j1),S,j2),N,a),I1,b))*R^j1*S^j2*N^a*I1^b:
od:
od:
od:
od:
mu:

end:


#qbin(q,n,k): the q-binomial coefficient
qbin:=proc(q,n,k) local i:
expand(normal(mul(1-q^(n-i),i=0..k-1)/mul(1-q^i,i=1..k))):
end:

#A(n,p,q): the g.f. of permutations according to (inv,maj)
A:=proc(n,p,q) local k,i:
option remember:
if n=0 then
 RETURN(1):
else
expand(add((-1)^(k-1)*qbin(p,n,k)*p^binomial(k,2)*mul(1-q^i,i=n-k+1..n-1)*
A(n-k,p,q),k=1..n)):
fi:
end:

#Bd(n,p,q): the g.f. of permutations according to (inv,maj)
#divided by n!*(p*q)^(binomial(n,2)/2)
Bd:=proc(n,p,q):
expand(A(n,p,q)/n!/(p*q)^(n*(n-1)/4)):
end:



#Bng(n,p,q): the g.f. of permutations according to (inv,maj)
#divided by n!*(p*q)^(binomial(n,2)/2)
Bng:=proc(n,p,q) local k,i:
option remember:
if n=0 then
 RETURN(1):
else
expand(add((-1)^(k-1)*qbin(p,n,k)*p^binomial(k,2)*mul(1-q^i,i=n-k+1..n-1)*
Bng(n-k,p,q)*(n-k)!/n!*p^(k*(3*k-1-2*n)/4)*q^(k*(k+1-2*n)/4)
,k=1..n)):
fi:
end:






#B(n,p,q): the g.f. of permutations according to (inv,maj)
B:=proc(n,p,q) local k,i:
option remember:
if n=0 then
 RETURN(1):
else
expand(add((-1)^(k-1)*qbin(p,n,k)*mul(1-q^i,i=n-k+1..n-1)*
B(n-k,p,q)*(n-k)!/n!*p^(k*(3*k-1-2*n)/4)*
q^(k*(-2*n+k+1)/4),k=1..n)):
fi:
end:





FindLeading:=proc(r,s,Aee,Aeo,Aoe,Aoo) local n,Fee,Feo,Foe,Foo,x:



#####

Fee:=Aee[r,s]*n^(3*(r+s)-3)*n^3-Aee[r,s]*(n-1)^(3*(r+s)-3)*(n-1)^3+
s*(n-1)*Aeo[r,s]*(n-1)^(3*(r+s)-3)-
s*(n-1)*Aeo[r,s]*(n-2)^(3*(r+s)-3):

Feo:=Aeo[r,s]*n^(3*r+3*s-3)-Aeo[r,s]*(n-1)^(3*r+3*s-3)
+(s-1/2)*(n-1)*(Aee[r,s-1]*(n-1)^(3*(r+s-1))- Aee[r,s-1]*(n-2)^(3*(r+s-1))):

Foe:=Aoe[r,s]*n^(3*r+3*s-4)*n-Aoe[r,s]*(n-1)^(3*r+3*s-4)*(n-1)+
s*(n-1)*(Aoo[r,s]*(n-1)^(3*r+3*s-4)-
         Aoo[r,s]*(n-2)^(3*r+3*s-4)):


Foo:=Aoo[r,s]*(n^(3*(r+s)-6)*n^2-(n-1)^(3*(r+s)-6)*(n-1)^2)
+(2*s-1)*(n-1)/2*Aoe[r,s-1]*((n-1)^(3*(r+s)-6)-(n-2)^(3*(r+s)-6)):


#####

Fee:=Aee[r,s]*n^3-Aee[r,s]*(1-x)^(3*(r+s)-3)*(n-1)^3+
s*(n-1)*Aeo[r,s]*(1-x)^(3*(r+s)-3)-
s*(n-1)*Aeo[r,s]*(1-2*x)^(3*(r+s)-3):

Feo:=Aeo[r,s]-Aeo[r,s]*(1-x)^(3*r+3*s-3)
+(s-1/2)*(n-1)*(Aee[r,s-1]*(1-x)^(3*(r+s-1))- Aee[r,s-1]*(1-2*x)^(3*(r+s-1))):

Foe:=Aoe[r,s]*n-Aoe[r,s]*(1-x)^(3*r+3*s-4)*(n-1)+
s*(n-1)*(Aoo[r,s]*(1-x)^(3*r+3*s-4)-
         Aoo[r,s]*(1-2*x)^(3*r+3*s-4)):


Foo:=Aoo[r,s]*(n^2-(1-x)^(3*(r+s)-6)*(n-1)^2)
+(2*s-1)*(n-1)/2*Aoe[r,s-1]*((1-x)^(3*(r+s)-6)-(1-2*x)^(3*(r+s)-6)):

Fee:=subs(n=1/x,Fee):
Feo:=subs(n=1/x,Feo):
Foe:=subs(n=1/x,Foe):
Foo:=subs(n=1/x,Foo):

series(Fee,x=0,7),
series(Feo,x=0,7),
series(Feo,x=0,7),
series(Foo,x=0,7):


end:



#Manig(P,x,y): the leading coefficient of the pol. P
Manig:=proc(P,x,y) local t,P1,d:
P1:=expand(subs({x=x*t,y=y*t},P)):
d:=degree(P1,t):
factor(coeff(P1,t,d)):
end:


FM8m:=proc(n,i) local gu:
gu:=FM8(n,i):
gu:=[seq([seq(Manig(gu[i1][j1],n,i),j1=1..nops(gu[i1]))],i1=1..nops(gu))]:

end:


FM8ee:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 1 to 4 do
for s from 1 to 4 do
 T[r,s]:=gu[2*r][2*s]/((2*r)!/r!/2^r)/((2*s)!/s!/2^s)*36^(r+s)/
n^(3*(r+s)):
od:
od:

evalb(normal({seq(seq(T[r,s],s=1..4),r=1..4)})={1}):

end:


FM8eo:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 1 to 4 do
for s from 1 to 4 do
 T[r,s]:=gu[2*r][2*s-1]/((2*r)!/r!/2^r)/((2*s)!/s!/2^s)*36^(r+s-1)/
(n^(3*(r+s-2))*(-(s-1)*n^3-6*r*n^2*i+18*r*n*i^2-12*r*i^3)):
od:
od:

evalb(normal({seq(seq(T[r,s],s=1..4),r=1..4)})={1}):


end:


FM8oe:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 2 to 4 do
for s from 2 to 4 do
 T[r,s]:=-gu[2*r-1][2*s]/((2*r)!/r!/2^r)/((2*s)!/s!/2^s)*36^(r+s)/36/
n^(3*(r+s-1))/(r-1):
od:
od:

evalb({seq(seq(normal(T[r,s]),s=2..4),r=2..4)}={1}):

end:




FM8oo:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 1 to 4 do
for s from 1 to 4 do
 T[r,s]:=gu[2*r-1][2*s-1]/((2*r)!/r!/2^r)/((2*s)!/s!/2^s)*36^(r+s)/
n^(3*(r+s)-6)/(162)/(2*i-n)^2:
od:
od:

evalb(normal({seq(seq(T[r,s],s=1..4),r=1..4)})={1}):
end:


GM8ee:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 1 to 4 do
for s from 1 to 4 do
 T[r,s]:=gu[2*r][2*s]-(2*r)!/r!/2^r*(2*s)!/s!/2^s*(1/36^(r+s))*n^(3*(r+s)):
od:
od:


evalb(normal({seq(seq(T[r,s],s=1..4),r=1..4)})={0}):
end:

GM8eo:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 1 to 4 do
for s from 1 to 4 do
 T[r,s]:=
expand(gu[2*r][2*s-1]-
(2*r)!/r!/2^r*(2*s)!/s!/2^s*(1/36)^(r+s-1)*
(n^(3*(r+s-2))*(-(s-1)*n^3-6*r*n^2*i+18*r*n*i^2-12*r*i^3))):
od:
od:

evalb(normal({seq(seq(T[r,s],s=1..4),r=1..4)})={0}):

end:


GM8oe:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 2 to 4 do
for s from 1 to 4 do
 T[r,s]:=
gu[2*r-1][2*s]
+
(2*r)!/r!/2^r*(2*s)!/s!/2^s*1/36^(r+s-1)*n^(3*(r+s-1))*(r-1):
od:
od:

evalb(normal({seq(seq(T[r,s],s=1..4),r=2..4)})={0}):

end:


GM8oo:=proc(n,i) local gu,T,r,s:
gu:=FM8m(n,i):
for r from 1 to 4 do
for s from 1 to 4 do
 T[r,s]:=expand(gu[2*r-1][2*s-1]-
(2*r)!/r!/2^r*(2*s)!/s!/2^s/36^(r+s-1)*9/2*n^(3*(r+s)-6)*(2*i-n)^2):
od:
od:

evalb(normal({seq(seq(T[r,s],s=1..4),r=1..4)})={0}):
end:



Bdokee:=proc() local r,s,n,EE,EO,OE,OO,i,guee,t1,t2,T:

#EE:=(2*r)!/r!/2^r*(2*s)!/s!/2^s*(1/36^(r+s))*n^(3*(r+s)):
EE:=(2*r)!/r!/2^r*(2*s)!/s!/2^s*(1/36^(r+s)):

#EO:=(2*r)!/r!/2^r*(2*s)!/s!/2^s*(1/36)^(r+s-1)*
#n^(3*(r+s-2))*(-(s-1)*n^3-6*r*n^2*i+18*r*n*i^2-12*r*i^3):

EO:=(2*r)!/r!/2^r*(2*s)!/s!/2^s*(1/36)^(r+s-1)*
(-(s-1)*n^3-6*r*n^2*i+18*r*n*i^2-12*r*i^3):

#OE:=-(2*r)!/r!/2^r*(2*s)!/s!/2^s*1/36^(r+s-1)*n^(3*(r+s-1))*(r-1):

OE:=-(2*r)!/r!/2^r*(2*s)!/s!/2^s*1/36^(r+s-1)*(r-1):

#OO:=(2*r)!/r!/2^r*(2*s)!/s!/2^s/36^(r+s-1)*9/2*n^(3*(r+s)-6)*(2*i-n)^2:
OO:=(2*r)!/r!/2^r*(2*s)!/s!/2^s/36^(r+s-1)*9/2*(2*i-n)^2:

#RETURN(EE,EO,OE,OO):

#1/n=t1 1/i=t2
guee:=1
-subs(n=1/t1,
 simplify(subs(i=i+1,EE)/EE
         )
    )
+2*s*t1^6*subs(n=1/t1,simplify(subs(i=i+1,EO)/EE))
-2*s*t1^6*(1-t1)^(3*(r+s-2))*subs(n=1/t1,simplify(subs(n=n-1,EO)/EE))
:


guee:=subs(i=1/t2,guee):
guee:=normal(subs({t1=t1*T,t2=t2*T},guee)):
guee:=taylor(guee,T=0,10):

evalb({seq(coeff(guee,T,i),i=0..3)}={0}):
end:






#Check1(L,n,i): applies the operator Oper1 to the list-of-lists L
Check1:=proc(L,n,i) local r,s,gu1,vu:

vu:={}:
for r from 1 to nops(L) do
for s from 2 to nops(L[r]) do

gu1:=L[r][s]-subs(i=i+1,L[r][s])+s*subs(i=i+1,L[r][s-1])-
s*subs(n=n-1,L[r][s-1]):
gu1:=expand(gu1):
vu:=vu union {degree(gu1,{n,i})-degree(L[r][s],{n,i})}:
od:
od:
evalb(max(op(vu))<0) :
end:


#Check2(L,n,i): applies the operator Oper2 to the list-of-lists L
Check2:=proc(L,n,i) local r,s,gu1,gu2,vu,lu:

vu:={}:
for r from 2 to nops(L) do
for s from 2 to nops(L[r]) do

gu1:=
normal(sum(subs(n=n-1,L[r][s]),i=1..n-1)/(n-1))-
normal(s/2*sum((2*i-n)*subs(n=n-1,L[r][s-1]),i=1..n-1)/(n-1))-
normal(r/2*sum((2*i-n)*subs(n=n-1,L[r-1][s]),i=1..n-1)/(n-1))
+
normal(r*s/4*sum((2*i-n)^2*subs(n=n-1,L[r-1][s-1]),i=1..n-1)/(n-1))
:
gu1:=expand(gu1):
gu2:=expand(subs(i=n,L[r][s])):
gu1:=gu2-gu1:
lu:=degree(gu1,{n,i})-degree(gu2,n):
  vu:=vu union {lu}:
od:
od:
evalb(max(op(vu))<0) :
end: