######################################################################
##P123: Save this file as  P123                                      #
## To use it, stay in the                                            #
##same directory, get into Maple (by typing: maple <Enter> )         #
##and then type:  read P123<Enter>                                   #
##Then follow the instructions given there                           #
##                                                                   #
##Written by Doron Zeilberger, Rutgers University ,                  #
#zeilberg at math dot rutgers dot edu                                #
######################################################################

print(`First Written: June-Aug., 2012: tested for Maple 15 `):
print(`Version of June-Aug. 2012  `):
print():
print(`This is P123, one of the  Maple packages`):
print(`accompanying Brian Nakamura and Doron Zeilberger's  article: `):
print(`"Using Noonan-Zeilberger Functional Equations `):
print(`to enumerate (in Polynomial Time!) Generalized Wilf classes"`):
print(`available from:`):
print(` http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/Gwilf.html .`):
print():
print(`The most current version of this program is available on WWW at:`):
print(` http://www.math.rutgers.edu/~zeilberg/tokhniot/P123 .`):
 


print(`-------------------------------------------------------------`):
 print(`For a list of the story procedures type ezraS();, for help with`):
 print(`a specific procedure, type ezraGuess(procedure_name);   .`):
 print(``):
print(`-------------------------------------------------------------`):

print(`-------------------------------------------------------------`):
 print(`For a list of the Guessing procedures type ezraGuess();, for help with`):
 print(`a specific procedure, type ezraGuess(procedure_name);   .`):
 print(``):
print(`-------------------------------------------------------------`):

print(`-------------------------------------------------------------`):
 print(`For a list of the Checking procedures type ezraC();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`-------------------------------------------------------------`):

print(`-------------------------------------------------------------`):
 print(`For a list of the supporting procedures type ezra1();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`-------------------------------------------------------------`):


print(`-------------------------------------------------------------`):
 print(`For a list of the MAIN procedures type ezra();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`-------------------------------------------------------------`):

ezraC:=proc()

if args=NULL then
 print(` The checking procedures are: CheckPn, PnDirect `):

else
ezra(args):
fi:

end:

ezraS:=proc()

if args=NULL then
 print(` The story procedures are: PCfV, Stat6V `):

else
ezra(args):
fi:

end:


ezraGuess:=proc()

if args=NULL then
 print(` The guessing procedures are: GuessPol, GuessRat, GuessRat1,`):
print(` GuessSeq, GuessSeqF,GuessSeqF1 `):
else
ezra(args):
fi:

end:

ezra1:=proc()

if args=NULL then
 print(` The supporting procedures are: AveAndMoms, Children `):
 print(`  LtoS  `):
 print(` PqLy, PqLyT, StoL, TrimPair, TrimPairs, Wt, Yeladim, YeladimT `):

else
ezra(args):
fi:

end:

ezra:=proc()

if args=NULL then
 print(`The main procedures are: ConjSeqs, ConjSeqsPC`):
 print(`ConjSeqsV, fn, L20, L236, PCf,`):
 print(` PqL, Pn, Seqs, SeqsPC, Stat6 `):
 print(` `):


elif nops([args])=1 and op(1,[args])=AveAndMoms then
print(`AveAndMoms(f,x,N): Given a probability generating function`):
print(`f(x) (a polynomial, rational function and possibly beyond)`):
print(`returns a list whose first entry is the average `):
print(`(alias expectation, alias mean)`):
print(`followed by the variance, the third moment (about the mean) and`):
print(`so on, until the N-th moment (about the mean).`):
print(`If f(1) is not 1, than it first normalizes it by dividing`):
print(`by f(1) (if it is not 0) .`):
print(`For example, try:`):
print(`AveAndMoms((1+x)^100,x,4);`):

elif nops([args])=1 and op(1,[args])=CheckPn then
print(`CheckPn(N):Checks Pn(n,q,y) for n from 1 to N `):
print(`Try: `):
print(` CheckPn(8); `):

elif nops([args])=1 and op(1,[args])=Children then
print(`Children(M,r): given a vector in short format, finds`):
print(`the set of its children also in short format`):
print(`try:`):
print(`Children([2,0,1],1);`):

elif nops([args])=1 and op(1,[args])=ConjSeqs then
print(`ConjSeqs(N,R,n): inputs a positive integer R and a symbol n`):
print(`and conjectures expressions for the sequences`):
print(`"number of permutations of length n with exacty r occurrences`):
print(`of the pattern 123" for r=0..,R as many as it can get. Try:`):
print(` ConjSeqs(20,3,n); `):

elif nops([args])=1 and op(1,[args])=ConjSeqsPC then
print(`ConjSeqsPC(N,R,n): Like ConjSeq(N,R,n)`):
print(`but if N<=25 and R<=7 it uses pre-computed values`):
print(`given by Seqs(N,R)`):
print(` Try:`):
print(` ConjSeqsPC(20,3,n); `):

elif nops([args])=1 and op(1,[args])=ConjSeqsV then
print(`ConjSeqsV(N,R,n): Verbose form of ConjSeqsV(N,R,n) `):
print(`Try: `):
print(` ConjSeqsV(20,3,n); `):

elif nops([args])=1 and op(1,[args])=fn then
print(`fn(n,q): the weight-enumerator of S_n according to the weight`):
print(`q^(#123)`):


elif nops([args])=1 and op(1,[args])=GuessPol then
print(`GuessPol(L,n): guesses a polynomial of degree d in n for`):
print(` the list L, such that P(i)=L[i] for i=1..nops(L) for example, try: `):
print(`GuessPol([seq(i,i=1..10)],n);`):

elif nops([args])=1 and op(1,[args])=GuessRat then
print(`GuessRat(L,n) guesses a rational function in n for the`):
print(`sequence L, try:`):
print(`GuessRat([seq((i^2+1)/(i^2+3),i=1..20)],n);`):

elif nops([args])=1 and op(1,[args])=GuessRat1 then
print(`GuessRat1(L,n,d) guesses a rational function in n for the`):
print(`sequence L of degree d. Try:`):
print(`GuessRat1([seq((i^2+1)/(i^2+3),i=1..20)],n,2);`):

elif nops([args])=1 and op(1,[args])=GuessSeq then
print(`GuessSeq(L,n): tries to guess an expression of the form`):
print(`binomial(2*n,n-beta)*RATIONAL(n) for the sequence L, `):
print(`Try:`):
print(`GuessSeq([seq((n1+5)*binomial(2*n1-1,n1-3),n1=1..20)],n);`):


elif nops([args])=1 and op(1,[args])=GuessSeqF then
print(`GuessSeqF(L,n): tries to guess an expression of the form`):
print(`(2*n-a)!/((n-beta)!*(n+c)!)*POL(n) for some positive integers`):
print(`a and c (a<=2*b) and c>=b both as small as possible`):
print(`the sequence L, try:`):
print(`where beta index in L that is not zero`):
print(`GuessSeqF([0$3,seq((n1+5)*(2*n1-3)!/((n1-3)!*(n1+5)!),n1=3..20)],n);`):

elif nops([args])=1 and op(1,[args])=GuessSeqF1 then
print(`GuessSeqF1(L,n,a,c): tries to guess an expression of the form`):
print(`(2*n-a)!/((n-beta)!*(n+c)!)*POL(n) for the sequence L, try:`):
print(`where beta index in L that is not zero`):
print(`GuessSeqF1([0$2,seq((n1+5)*(2*n1-3)!/((n1-3)!*(n1+5)!),n1=3..20)],n,3,5);`):

elif nops([args])=1 and op(1,[args])=L20 then
print(`L20(q): the  precomputed [seq(fn(n1,q),n1=1..20)]. Try: `):
print(`L20(q);`):

elif nops([args])=1 and op(1,[args])=L236 then
print(`L236(q): the pre-computed truncated up to q^6 of the`):
print(`weight-enumerator by q^#123(pi) of S_n, try`):
print(`L236(q) `):

elif nops([args])=1 and op(1,[args])=LtoS then
print(`LtoS(L,q,r): given a vector of increasing powers of q up to the`):
print(`q^(r+1) returns the list of integers of length r+2 whose`):
print(`j-th entry is the number of times q^j shows up`):
print(`Try LtoS([1,1,q,q,q^3,q^3,q^3],q,2);`):


elif nops([args])=1 and op(1,[args])=PCf then
print(`PCf(n):the list`):
print(`whose (r+1)-th entry is the`):
print(`precomputed explicit (conjectured) expressions`):
print(`for the number of 123 n-permutations with exactly r`):
print(`occurrences of 123. Try:`):
print(`PCf(n);`):

elif nops([args])=1 and op(1,[args])=PCfV then
print(`PCfV(n): verbose form of PCf(n) `):
print(`Try:`):
print(`PCfV(n);`):

elif nops([args])=1 and op(1,[args])=Pn then
print(`Pn(n,q,y):The weight-enumerator of perms of {1, ..., n}`):
print(`according to the weight q#123*y[1]#(12,i=1)*y[2]#(12,1=2) ...`):
print(`try: `):
print(`Pn(5,q,y); `):

elif nops([args])=1 and op(1,[args])=PnDirect then
print(`PnDirect(n,q,y):The weight-enumerator of perms of {1, ..., n}`):
print(`according to the weight q#123*y[1]#(12,i=1)*y[2]#(12,1=2) ...`):
print(`computed directly. FOR CHECKING PURPOSES ONLY, VERY SLOW!`):
print(`try: `):
print(`PnDirect(5,q,y); `):

elif nops([args])=1 and op(1,[args])=PqL then
print(`PqL(q,L):The value of P_n(n,q,y) where L=nops(L)`):
print(`and one plugs in for [y[1], ..., y[n]] the`):
print(`values of L, try:`):
print(`PqL(q,[y1,y2,y3]);`):

elif nops([args])=1 and op(1,[args])=PqLy then
print(`Like PqL(q,L), but using Yeladim(L,q)`):
print(`Try:`):
print(`PqLy(q,[y1,y2,y3]);`):

elif nops([args])=1 and op(1,[args])=PqLyT then
print(`PqLyT(q,L,r):Like PqLy but only interested in terms up`):
print(`to the r-th power of q`):
print(`Try:`):
print(`PqLyT(q,[1,1,1,1],1);`):


elif nops([args])=1 and op(1,[args])=Seqs then
print(`Seqs(N,R): The lists whose R+1 lists whose (r+1)-th list`):
print(`is the first N members (starting at n=1)`):
print(`number of permutations with exactly r occurrence of 123,`):
print(`try: Seqs(8,3);`):

elif nops([args])=1 and op(1,[args])=SeqsPC then
print(`SeqsPC(N,R): The pre-computed Seqs(N,R) for N<=25 and R<=7`):
print(`try: SeqsPC(8,3);`):

elif nops([args])=1 and op(1,[args])=Stat6 then
print(`Stat6(n): the expectation, variance, and the 3rd-6th moments`):
print(`about the mean of the random variable #occurrences `):
print(`of the pattern 123 in S_n. `):
print(`Also the alpha coefficients up to the sixth`):
print(`(for the odd ones you get the square)`):
print(`Try:`):
print(`Stat6(n);`):

elif nops([args])=1 and op(1,[args])=Stat6V then
print(`Stat6V(n): Verbose version of Stat6(n) (q.v.) `):
print(`Try:`):
print(`Stat6V(n);`):

elif nops([args])=1 and op(1,[args])=StoL then
print(`StoL(M,q,r): given a vector in short format`):
print(`converts it to long format, try:`):
print(`StoL([2,1,3],q,1);`):

elif nops([args])=1 and op(1,[args])=TrimPair then
print(`TrimPair(P,q,r): Given a Pair P=[coeff,L]`):
print(`where coeff is a power of q, `):
print(` and L is a list of powers of q, returns the `):
print(`empty set if coeff has power larger than r,`):
print(`and otherwise returns the singleton`):
print(`{[coeff,L1]} where L1 is L with`):
print(`any power larger than q^(r+1) is replaced by q^(r+1)`):
print(`For example, try: `):
print(`TrimPair([q,[1,1,q,q,q^3,q^4,q^4]],q,2);`):

elif nops([args])=1 and op(1,[args])=TrimPairs then
print(`TrimPairs(S,q,r): the set of trimmed pairs coming from the`):
print(`set of pairs S with respect to the power r, try:`):
print(`TrimPairs({[q,[1,1,q,q,q^3,q^4,q^4]]},q,1);`):

elif nops([args])=1 and op(1,[args])=Wt then
print(`Wt(pi,q,y): the weight of a permutation pi according to`):
print(`q^(#123)*y[1]#(12,i=1)*y[2]#(12,1=2) ... `):
print(`Try: `):
print(`Wt([1,2,3,4],q,y); `):



elif nops([args])=1 and op(1,[args])=Yeladim then
print(`Yeladim(L,q): the children of the list L`):
print(`Try: Yeladim([1,1,1],q);`):

elif nops([args])=1 and op(1,[args])=YeladimT then
print(`YeladimT(L,q,r): the trimmed children of the list L`):
print(`with respect to the power r.`):
print(`Try: YeladimT([1,q^3,q^5],q,2);`):

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


#L20(q): the  precomputed [seq(fn(n1,q),n1=1..20)]. Try: 
#L20(q);
L20:=proc(q):
[1, 2, 5+q, 3*q^2+q^4+6*q+14, 42+24*q^2+7*q^3+4*q^7+6*q^5+q^10+9*q^4+27*q, 132+
133*q^2+70*q^3+32*q^7+54*q^5+16*q^10+37*q^6+24*q^8+6*q^12+5*q^16+12*q^9+8*q^13+
74*q^4+6*q^11+q^20+110*q, 429+635*q^2+461*q^3+320*q^7+395*q^5+191*q^10+387*q^6+
260*q^8+104*q^12+74*q^16+232*q^9+130*q^13+507*q^4+10*q^25+6*q^30+30*q^21+18*q^
18+10*q^26+162*q^11+10*q^20+62*q^17+13*q^22+100*q^14+32*q^19+24*q^15+8*q^23+q^
35+429*q, 1430+2807*q^2+2528*q^3+2544*q^7+2570*q^5+2077*q^10+2864*q^6+2389*q^8+
1580*q^12+1047*q^16+2182*q^9+1456*q^13+3008*q^4+324*q^25+154*q^30+466*q^21+682*
q^18+252*q^26+1818*q^11+546*q^20+1004*q^17+537*q^22+1494*q^14+656*q^19+886*q^15
+288*q^23+38*q^35+115*q^28+68*q^34+16*q^40+138*q^29+228*q^24+47*q^36+66*q^31+
156*q^27+60*q^33+58*q^32+18*q^38+15*q^44+7*q^50+40*q^39+12*q^45+10*q^41+q^56+
1638*q, 4862+11864*q^2+12525*q^3+17357*q^7+15203*q^5+17505*q^10+18179*q^6+18096
*q^8+15847*q^12+12472*q^16+17333*q^9+15068*q^13+16151*q^4+6072*q^25+3412*q^30+
8225*q^21+10837*q^18+5158*q^26+16605*q^11+8935*q^20+12101*q^17+8089*q^22+15049*
q^14+9588*q^19+12630*q^15+6836*q^23+1995*q^35+3901*q^28+1887*q^34+850*q^40+3462
*q^29+5405*q^24+1583*q^36+2976*q^31+4541*q^27+2151*q^33+2524*q^32+1064*q^38+488
*q^44+166*q^50+1190*q^39+420*q^45+834*q^41+57*q^56+508*q^43+1312*q^37+282*q^48+
458*q^46+234*q^51+427*q^47+19*q^65+110*q^58+148*q^52+823*q^42+50*q^60+248*q^53+
186*q^49+44*q^54+66*q^57+80*q^55+20*q^63+58*q^59+8*q^77+21*q^70+60*q^64+14*q^71
+12*q^66+10*q^62+q^84+6188*q, 16796+48756*q^2+58258*q^3+107194*q^7+83382*q^5+
129449*q^10+105082*q^6+120197*q^8+132579*q^12+127049*q^16+121630*q^9+130310*q^
13+80889*q^4+84020*q^25+61259*q^30+103134*q^21+121330*q^18+82579*q^26+128712*q^
11+109115*q^20+121602*q^17+102875*q^22+133945*q^14+112240*q^19+124572*q^15+
95256*q^23+41368*q^35+69645*q^28+44615*q^34+27361*q^40+64590*q^29+86372*q^24+
39552*q^36+56574*q^31+73540*q^27+47734*q^33+53742*q^32+31990*q^38+18612*q^44+
9812*q^50+29182*q^39+16776*q^45+25088*q^41+5451*q^56+20574*q^43+35538*q^37+
14304*q^48+15679*q^46+9188*q^51+14582*q^47+2434*q^65+5058*q^58+8974*q^52+23871*
q^42+3486*q^60+9012*q^53+11058*q^49+6788*q^54+4708*q^57+5824*q^55+2206*q^63+
4682*q^59+700*q^77+1480*q^70+2678*q^64+1130*q^71+1977*q^66+2795*q^62+326*q^84+
986*q^72+192*q^80+1256*q^67+608*q^75+1192*q^68+3004*q^61+401*q^74+1134*q^69+296
*q^83+603*q^76+264*q^82+172*q^90+22*q^98+59*q^92+60*q^85+312*q^79+378*q^78+64*q
^87+606*q^73+72*q^91+107*q^86+156*q^81+120*q^89+15*q^88+84*q^97+35*q^96+28*q^
104+9*q^112+16*q^105+14*q^99+12*q^94+q^120+23256*q, 58786+196707*q^2+259787*q^3
+616449*q^7+431167*q^5+868607*q^10+568809*q^6+730182*q^8+975820*q^12+1099257*q^
16+778432*q^9+1000220*q^13+385387*q^4+1005574*q^25+867777*q^30+1079286*q^21+
1121794*q^18+1002920*q^26+901457*q^11+1106169*q^20+1092656*q^17+1094214*q^22+
1059437*q^14+1101327*q^19+1051194*q^15+1065888*q^23+694925*q^35+925079*q^28+
735291*q^34+536582*q^40+897540*q^29+1031696*q^24+665451*q^36+833036*q^31+963317
*q^27+761461*q^33+807892*q^32+601738*q^38+428537*q^44+288746*q^50+567469*q^39+
402525*q^45+514598*q^41+191347*q^56+457925*q^43+644967*q^37+343753*q^48+377705*
q^46+273107*q^51+354636*q^47+98752*q^65+167509*q^58+250591*q^52+487819*q^42+
140646*q^60+248644*q^53+315180*q^49+223830*q^54+177756*q^57+206056*q^55+111283*
q^63+159312*q^59+35811*q^77+65623*q^70+105226*q^64+60868*q^71+89678*q^66+124559
*q^62+19063*q^84+54635*q^72+27264*q^80+84846*q^67+42730*q^75+74486*q^68+133790*
q^61+43978*q^74+68371*q^69+20715*q^83+39349*q^76+21425*q^82+11210*q^90+4878*q^
98+8278*q^92+16286*q^85+28528*q^79+32045*q^78+15964*q^87+50256*q^73+8690*q^91+
14829*q^86+25732*q^81+11168*q^89+11766*q^88+4362*q^97+5607*q^96+1787*q^104+398*
q^112+1862*q^105+3660*q^99+7356*q^94+150*q^120+6657*q^95+4146*q^101+8326*q^93+
3456*q^102+1310*q^110+166*q^119+746*q^113+3955*q^100+1806*q^108+1688*q^106+2283
*q^107+1089*q^115+850*q^116+533*q^122+867*q^114+392*q^123+224*q^131+25*q^140+
534*q^117+2574*q^103+1756*q^109+462*q^124+68*q^133+508*q^118+64*q^125+874*q^111
+66*q^127+166*q^121+148*q^126+35*q^129+98*q^132+56*q^138+222*q^130+112*q^139+36
*q^147+10*q^156+18*q^148+16*q^141+14*q^135+q^165+87210*q, 208012+783750*q^2+
1124704*q^3+3357324*q^7+2129734*q^5+5409361*q^10+2930602*q^6+4158993*q^8+
6582534*q^12+8554680*q^16+4649860*q^9+7005086*q^13+1769705*q^4+10363514*q^25+
10298618*q^30+9887754*q^21+9302926*q^18+10534421*q^26+5850192*q^11+9848663*q^20
+8800138*q^17+10252212*q^22+7664290*q^14+9455254*q^19+7955082*q^15+10289744*q^
23+9411088*q^35+10439795*q^28+9696595*q^34+8326384*q^40+10286752*q^29+10412901*
q^24+9248159*q^36+10083848*q^31+10461734*q^27+9782454*q^33+10060255*q^32+
8858488*q^38+7351731*q^44+5840485*q^50+8529828*q^39+7038964*q^45+8050836*q^41+
4459522*q^56+7566114*q^43+9051636*q^37+6353104*q^48+6825097*q^46+5557854*q^51+
6520322*q^47+2809980*q^65+4065748*q^58+5315981*q^52+7878546*q^42+3655541*q^60+
5124276*q^53+6064454*q^49+4931245*q^54+4237332*q^57+4648480*q^55+3128488*q^63+
3856570*q^59+1410444*q^77+2143602*q^70+2987967*q^64+2019970*q^71+2673316*q^66+
3339915*q^62+905971*q^84+1907813*q^72+1170509*q^80+2536156*q^67+1591784*q^75+
2404584*q^68+3476616*q^61+1686055*q^74+2249478*q^69+964510*q^83+1520651*q^76+
1036376*q^82+616455*q^90+343986*q^98+514716*q^92+831612*q^85+1226822*q^79+
1317057*q^78+747260*q^87+1781342*q^73+557190*q^91+789574*q^86+1106488*q^81+
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*q^916+23044706*q^899+2862672*q^949+206540*q^981+448310*q^964+3248*q^1025+
106630*q^990+405076*q^973+23088*q^1007+523564*q^959+12057*q^1008+673956*q^946+
141014*q^975+197634*q^963+617554*q^958+35998*q^992+93134*q^977+30*q^1060+
9517294*q^915+1471728*q^966+1790594*q^948+55480*q^991+528*q^1026+176*q^1043+
77557*q^998+36*q^1105+149*q^1072+2910*q^1054+192*q^1030+448*q^1013+5676*q^1009+
156*q^1057+5358*q^1023+32*q^1074]:
end:



#L236(q): the pre-computed truncated up to q^6 of the
#weight-enumerator by q^#123(pi) of S_n, tryL
#L236(q)
L236:=proc(q):
[1, 2, 5+q, 3*q^2+q^4+6*q+14, 42+27*q+24*q^2+7*q^3+9*q^4+6*q^5, 132+110*q+133*q
^2+70*q^3+74*q^4+54*q^5+37*q^6, 429+429*q+635*q^2+461*q^3+507*q^4+395*q^5+387*q
^6, 1430+1638*q+2807*q^2+2528*q^3+3008*q^4+2570*q^5+2864*q^6, 4862+6188*q+11864
*q^2+12525*q^3+16151*q^4+15203*q^5+18179*q^6, 16796+23256*q+48756*q^2+58258*q^3
+80889*q^4+83382*q^5+105082*q^6, 58786+87210*q+196707*q^2+259787*q^3+385387*q^4
+431167*q^5+568809*q^6, 208012+326876*q+783750*q^2+1124704*q^3+1769705*q^4+
2129734*q^5+2930602*q^6, 742900+1225785*q+3095708*q^2+4765761*q^3+7902982*q^4+
10148146*q^5+14527238*q^6, 2674440+4601610*q+12152855*q^2+19873150*q^3+34539352
*q^4+46993138*q^5+69819523*q^6, 9694845+17298645*q+47500635*q^2+81864705*q^3+
148409950*q^4+212661944*q^5+327200581*q^6, 35357670+65132550*q+185082495*q^2+
334052160*q^3+629096364*q^4+944504744*q^5+1501719377*q^6, 129644790+245642760*q
+719559600*q^2+1353003990*q^3+2637532977*q^4+4130591771*q^5+6773007550*q^6, 
477638700+927983760*q+2793121080*q^2+5447702664*q^3+10958971820*q^4+17833668406
*q^5+30100185693*q^6, 1767263190+3511574910*q+10830450780*q^2+21830542860*q^3+
45197027220*q^4+76170163860*q^5+132099138291*q^6, 6564120420+13309856820*q+
41965864794*q^2+87145866752*q^3+185248724362*q^4+322377660872*q^5+573518305776*
q^6, 24466267020+50528160150*q+162539516448*q^2+346793185822*q^3+755330691518*q
^4+1353835501123*q^5+2466880451752*q^6, 91482563640+192113383644*q+629399492330
*q^2+1376521109436*q^3+3066235194908*q^4+5647648935466*q^5+10525239411665*q^6,
343059613650+731508653106*q+2437072038302*q^2+5452332482690*q^3+12400715391073*
q^4+23424344330803*q^5+44590535671158*q^6]:
end:


#PCf(n):the list
#whose (r+1)-th entry is the
#precomputed explicit (conjectured) expressions
#for the number of 123 n-permutations with exactly r
#occurrences of 123. Try:
#PCf(n);
PCf:=proc(n):

[2*(2*n-1)!/(n-1)!/(n+1)!, 6*(2*n-1)!/(n-3)!/(n+3)!, (100+117*n+59*n^2)*(2*n-2)!/(n-4)!/(n+5)!, 4*n*(1804+937*n+506*n^2+113*n^3)*(2*n-3)!/(n-5)!/(n+7)!, (-\
39191040-46806*n^6+26283456*n+3126*n^7-659091*n^4+5520576*n^2+3561*n^8+2630634*n^3+12384*n^5)*(2*n-4)!/(n-4)!/(n+9)!, (-32891443200-9648450*n^6+3611692800*n+
22524*n^7+381205612*n^4+2800103664*n^2-115872*n^8+1607633896*n^3+71304534*n^5+136646*n^9+26246*n^10)*(2*n-5)!/(n-5)!/(n+11)!, (-28177269120000+14335663329*n^6-\
6909513045120*n+2889552582*n^7+193311*n^12+313010684796*n^4+1283595375168*n^2+406430793*n^8+1025692693464*n^3+60005854890*n^5+2349954*n^11+95151030*n^9+13035003*
n^10)*(2*n-6)!/(n-6)!/(n+13)!, (-763734137886720000-86395090065440*n^6-157858417817856000*n+355187760*n^13-26631101348520*n^7+22526480*n^14-1654450096*n^12+
3647384624274048*n^4+103032675524966400*n^2+13111080*n^15+3750695521216*n^8+11386434230674560*n^3-636425872408320*n^5+10534951680*n^11-305671694640*n^9+
15797223760*n^10+1386032*n^16)*(2*n-7)!/(n-5)!/(n+15)!]:

end:


#Pn(n,q,y):The weight-enumerator of perms of {1, ..., n}
#according to the weight q#123*y[1]#(12,i=1)*y[2]#(12,1=2) ...
Pn:=proc(n,q,y) local i,j,gu:
option remember:
if n=1 then
 RETURN(1):
else
 gu:=Pn(n-1,q,y):
 expand(add(y[i]^(n-i)*subs({seq(y[j]=q*y[j+1],j=i..n-2)},gu),i=1..n)):
fi:
end:


#PnListOld(n,q,y):The weight-enumerator of perms of {1, ..., n}
#according to the weight q#123*y[1]#(12,i=1)*y[2]#(12,1=2) ...
PnList:=proc(n,q,y) local i,j,gu:
option remember:
if n=1 then
 RETURN([1]):
else
 gu:=Pn(n-1,q,y):
RETURN(
[seq(
expand(y[i]^(n-i)*subs({seq(y[j]=q*y[j+1],j=i..n-1)},gu)),
i=1..n)]):
fi:
gu:
end:


#fn(n,q): the weight-enumerator of S_n according to the weight
#q^(#123)
fn:=proc(n,q) local gu,i,y:
gu:=Pn(n,q,y):
sort(subs({seq(y[i]=1,i=1..n)},gu)):
end:


#SeqsOld(N,R): The lists whose R+1 lists whose (r+1)-th list
#is the first N members (starting at n=1)
#number of permutations with exactly r occurrence of 123,
#try: Seqs(8,7);
SeqsOld:=proc(N,R) local gu,n1,q,r:
gu:=[seq(fn(n1,q),n1=1..N)]:
[seq([seq(coeff(gu[n1],q,r),n1=1..N)],r=0..R)]:
end:



#PnT(n,q,y,R):The weight-enumerator of perms of {1, ..., n}
#according to the weight q#123*y[1]#(12,i=1)*y[2]#(12,1=2) ...
#truncated to the first R powers of q, Try:
PnT(3,q,y,2);
PnT:=proc(n,q,y,R) local i,j,gu,mu:
option remember:
if n=1 then
 RETURN(1):
else
 gu:=PnT(n-1,q,y,R):
mu:= expand(add(y[i]^(n-i)*subs({seq(y[j]=q*y[j+1],j=i..n-2)},gu),i=1..n)):
mu:=add(coeff(mu,q,i)*q^i,i=0..R):
fi:
end:


#fnT(n,q,R): the weight-enumerator of S_n according to the weight
#q^(#123) truncated to q^R
fnT:=proc(n,q,R) local gu,i,y:
gu:=PnT(n,q,y,R):
sort(subs({seq(y[i]=1,i=1..n)},gu)):
end:



#SeqsSlow(N,R): The lists whose R+1 lists whose (r+1)-th list
#is the first N members (starting at n=1)
#number of permutations with exactly r occurrence of 123,
#try: SeqsSlow(8,7);
SeqsSlow:=proc(N,R) local gu,n1,q,r:
gu:=[seq(fnT(n1,q,R),n1=1..N)]:
[seq([seq(coeff(gu[n1],q,r),n1=1..N)],r=0..R)]:
end:



#PqL(q,L):The value of P_n(n,q,y) where L=nops(L)
#and one plugs in for [y[1], ..., y[n]] the
#values of L, try:
#PqL(q,[y1,y2,y3]);
PqL:=proc(q,L) local n,i,j:
option remember:
n:=nops(L):
if n=1 then
 RETURN(1):
else
expand(
add(L[i]^(n-i)*PqL(q,[op(1..i-1,L),seq(q*L[j],j=i+1..n)]),
i=1..n)):
fi:
end:



#PqLt(q,L,r):The value of P_n(n,q,y) where L=nops(L)
#and one plugs in for [y[1], ..., y[n]] the
#values of L, try: truncated up to the r-th power of q
#PqL(q,[y1,y2,y3]);
PqLt:=proc(q,L,r) local n,i,j,lu:
option remember:
n:=nops(L):
if n=1 then
 RETURN(1):
else
lu:=expand(
add(L[i]^(n-i)*PqLt(q,[op(1..i-1,L),seq(q*L[j],j=i+1..n)],r),
i=1..n)):
lu:=add(coeff(lu,q,i)*q^i,i=0..r):
RETURN(lu):
fi:

end:

#Yeladim(L,q): the children of the list L
#Try: Yeladim([1,1,1],q);
Yeladim:=proc(L,q) local n,i,j:
n:=nops(L):
[seq([L[i]^(n-i),[op(1..i-1,L),seq(q*L[j],j=i+1..n)]],i=1..n)];
end:


#PqLy(q,L):Like PqL but using Yeladim
#Try:
#PqLy(q,[y1,y2,y3]);
PqLy:=proc(q,L) local n,i,gu:
option remember:
n:=nops(L):
if n=1 then
 RETURN(1):
else
gu:=Yeladim(L,q):
RETURN(expand(add(gu[i][1]*PqLy(q,gu[i][2]),i=1..nops(gu)))):
fi:
end:

#TrimPair(P,q,r): Given a Pair P=[coeff,L]
#where coeff is a power of q, 
#and L is a list of powers of q, returns the
#empty set if coeff has power larger than r,
#and otherwise returns the singleton
#{[coeff,L1]} where L1 is L with
#any power larger than q^(r+1) is replaced by q^(r+1)
#For example, try: 
#TrimPair([q,[1,1,q,q,q^3,q^4,q^4]],q,2);
TrimPair:=proc(P,q,r) local c,L,L1,i:
c:=P[1]:
L:=P[2]:
if degree(c,q)>r then
 RETURN({}):
fi:

L1:=[]:

for i from 1 to nops(L) do
 if degree(L[i],q)>r+1 then
  L1:=[op(L1),q^(r+1)]:
 else
  L1:=[op(L1),L[i]]:
 fi:
od:
{[c,L1]}:
end:



#TrimPairs(S,q,r): the set of trimmed pairs coming from the
#set of pairs S with respect to the power r, try:
#TrimPairs({[q,[1,1,q,q,q^3,q^4,q^4]]},q,1);
TrimPairs:=proc(S,q,r) local P:
{seq(op(TrimPair(P,q,r)), P in S)}:
end:



#YeladimT(L,q,r): the trimmed children of the list L
#with respect to the power r.
#Try: YeladimT([1,q^3,q^5],q,2);
YeladimT:=proc(L,q,r):
TrimPairs(Yeladim(L,q),q,r):
end:


#PqLyT(q,L,r):Like PqLy but only interested in terms up
#to the r-th power of q
#Try:
#PqLyT(q,[1,1,1,1],1);
PqLyT:=proc(q,L,r) local n,i,gu,mu:
option remember:
n:=nops(L):
if n=1 then
 RETURN(1):
else
gu:=YeladimT(L,q,r):
mu:=expand(add(gu[i][1]*PqLyT(q,gu[i][2],r),i=1..nops(gu))):
mu:=add(coeff(mu,q,i)*q^i,i=0..r):
RETURN(mu):
fi:
end:

#Seqs(N,R): The lists whose R+1 lists whose (r+1)-th list
#is the first N members (starting at n=1)
#number of permutations with exactly r occurrence of 123,
#try: Seqs(8,3);
Seqs:=proc(N,R) local gu,n1,q,r:
option remember:
gu:=[seq(PqLyT(q,[1$n1],R),n1=1..N)]:
[seq([seq(coeff(gu[n1],q,r),n1=1..N)],r=0..R)]:
end:

#GuessRat1(L,n,d) guesses a rational function in n for the
#sequence L, try:
#GuessRat1([seq((i^2+1)/(i^2+3)),i=1..20)],n,2);
GuessRat1:=proc(L,n,d) local eq,var,a,b,R,Top,Bot,var1,i:
if nops(L)<2*d+6 then
 print(`list too small`):
 RETURN(FAIL):
fi:

var:={seq(a[i],i=0..d),seq(b[i],i=0..d)}:
Top:=add(a[i]*n^i,i=0..d):
Bot:=add(b[i]*n^i,i=0..d):
R:=Top/Bot:
eq:={seq(numer(L[i]-subs(n=i,R)),i=1..nops(L))}:

var1:=solve(eq,var):

Top:=subs(var1,Top):

if Top=0 then
 RETURN(FAIL):
fi:

Bot:=subs(var1,Bot):

normal(Top/Bot):

end:





#GuessRat(L,n) guesses a rational function in n for the
#sequence L, try:
#GuessRat([seq((i^2+1)/(i^2+3)),i=1..20)],n);
GuessRat:=proc(L,n) local d,gu:

for d from 0 to trunc(nops(L)/2)-3 do
 gu:=GuessRat1(L,n,d):
 if gu<>FAIL then
   RETURN(gu):
 fi:
od:
FAIL:
end:


#ConjSeqs(N,R,n): inputs a positive integer R and a symbol n
#and conjectures expressions for the sequences
#"number of permutations of length n with exacty r occurrences
#of the pattern r" for r=0..,R as many as it can get. Try:
#ConjSeqs(20,3,n);
ConjSeqs:=proc(N,R,n) local lu,gu,i,lu1,gu1:
lu:=Seqs(N,R):
gu:=[]:

for i from 0 to R do
lu1:=lu[i+1]:
gu1:=GuessSeq(lu1,n):
 if gu1=FAIL then
   RETURN(gu):
 else
  gu:=[op(gu),gu1]:
 fi:
od:

gu:

end:



#ConjSeqsPC(N,R,n): Like 
#ConjSeqs(N,R,n): but using SeqsPC(N,R):
#Try:
#ConjSeqsPC(20,3,n);
ConjSeqsPC:=proc(N,R,n) local lu,gu,i,lu1,gu1:
lu:=SeqsPC(N,R):
if lu=FAIL then
 RETURN(FAIL):
fi:

gu:=[]:

for i from 0 to R do
lu1:=lu[i+1]:
gu1:=GuessSeq(lu1,n):
 if gu1=FAIL then
   RETURN(gu):
 else
  gu:=[op(gu),gu1]:
 fi:
od:

gu:

end:





#ConjSeqsV(N,R,n): Verbose form of
#ConjSeqsV(N,R,n): 
#Try:
#ConjSeqsV(20,3,n);
ConjSeqsV:=proc(N,R,n) local gu,i,t0:
t0:=time():
gu:=ConjSeqs(N,R,n):


if gu=[] then
 RETURN(FAIL):
fi:

print(`Conjectures for the Number of Permutations of Length n`):
print(`With exactly r occurrences of the pattern 123`):
print(`for r between 0 and `, nops(gu)-1 ):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
for i from 1 to nops(gu) do
print(`The Number of permutations of length n with exactly`, i-1):
print(` occurrences  of the pattern 123 is`):
print(gu[i]):
print(``):
print(`and in Maple input form it is:`):
lprint(gu[i]):
od:

print(`This took`, time()-t0, `seconds. `):

end:




#LtoS(L,q,r): given a vector of increasing powers of q up to the
#q^(r+1) returns the list of integers of length r+2 whose
#j-th entry is the number of times q^j shows up
#Try LtoS([1,1,q,q,q^3,q^3,q^3],q,2);
LtoS:=proc(L,q,r) local L1,i,x,gu:

L1:=[seq(degree(L[i],q),i=1..nops(L))]:
if max(op(L1))>r+1 then
 RETURN(FAIL):
fi:

if sort(L1)<>L1 then
 RETURN(FAIL):
fi:

gu:=add(x[L1[i]],i=1..nops(L1)):
[seq(coeff(gu,x[i],1),i=0..r+1)]:
end:


#StoL(M,q): given a vector in short format
#converts it to long format, try:
#StoL([2,1,3],q,1);
StoL:=proc(M,q,r) local i:
if nops(M)<>r+2 then
RETURN(FAIL):
fi:
[seq((q^(i-1))$M[i],i=1..nops(M))]:
end:


#Children(M,r): given a vector in short format, finds
#the set of its children also in short format
#try:
#Children([2,0,1],1);
Children:=proc(M,r) local q,L,gu,i:
L:=StoL(M,q,r):

gu:=YeladimT(L,q,r):

[seq([degree(gu[i][1],q),LtoS(gu[i][2],q,r)],i=1..nops(gu))]:

end:






##begin guessing polynomials
#GuessPol1(L,d,n): guesses a polynomial of degree d in n for
# the list L, such that P(i)=L[i+1] for i=1..nops(L)
#For example, try: 
#GuessPol1([seq(i,i=1..10)],1,n);
GuessPol1:=proc(L,d,n) local P,i,a,eq,var:
if d>nops(L)-2 then
 ERROR(`the list is too small`):
fi:

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

var:=solve(eq,var):

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

subs(var,P):

end:

#GuessPol(L,n): guesses a polynomial of degree d in n for
# the list L, such that P(i)=L[i] for i=1..nops(L) for example, try: 
#GuessPol([seq(i,i=1..10)],n);
GuessPol:=proc(L,n) local d,gu:

for d from 0 to nops(L)-2 do
 gu:=GuessPol1(L,d,n):
 if gu<>FAIL then
    RETURN(gu):
 fi:
od:

FAIL:

end:
##end guessing polynomials








#GuessSeqF1(L,n,a,c): tries to guess an expression of the form
#(2*n-a)!/((n-beta)!*(n+c)!)*POL(n) for the sequence L, try:
#where beta index in L that is not zero
#GuessSeqF1([0$3,seq((n1+5)*(2*n1-3)!/((n1-3)!*(n1+5)!),n1=3..20)],n,3,5);
GuessSeqF1:=proc(L,n,a,c) local L1,b,i,Po:
for b from 1 to nops(L) while L[b]=0 do od:
b:
L1:=[seq(L[i]/(2*i-a)!*(i-b)!*(i+c)!,i=b..nops(L))]:

Po:=GuessPol(L1,n):

if Po=FAIL then
 RETURN(FAIL):
fi:
Po:=factor(subs(n=n-b+1,Po)):
Po:=Po*(2*n-a)!/((n-b)!*(n+c)!):

if [seq(eval(subs(n=i,Po)),i=b..nops(L))]<>[op(b..nops(L),L)] then
 print(Po, `did not work out`):
 print([seq(subs(n=i,Po),i=1..nops(L))],L):
 RETURN(FAIL):
else
 RETURN(Po):
fi:

end:



#GuessSeq(L,n): tries to guess an expression of the form
#(2*n-a)!/((n-beta)!*(n+c)!)*POL(n) for some positive integers
#a and c (a<=2*b) and c>=b both as small as possible
#the sequence L, try:
#where beta index in L that is not zero
#GuessSeq([0$3,seq((n1+5)*(2*n1-3)!/((n1-3)!*(n1+5)!),n1=3..20)],n);
GuessSeq:=proc(L,n) local b,gu,a,c:

for b from 1 to nops(L) while L[b]=0 do od:
b:


for c from  b to 4*b do
for a from 0 to 2*b do
gu:=GuessSeqF1(L,n,a,c):
if gu<>FAIL then
 RETURN(gu):
fi:
od:
od:
FAIL:

end:










#AveAndMoms(f,x,N): Given a probability generating function
#f(x) (a polynomial, rational function and possibly beyond)
#returns a list whose first entry is the average 
#(alias expectation, alias mean)
#followed by the variance, the third moment (about the mean) and
#so on, until the N-th moment (about the mean).
#If f(1) is not 1, than it first normalizes it by dividing
#by f(1) (if it is not 0) .
#For example, try:
#AveAndMoms(((1+x)/2)^100,x,4);

AveAndMoms:=proc(f,x,N) local mu,F,memu1,gu,i:
mu:=simplify(subs(x=1,f)):

if mu=0 then
print(f, `is neither a prob. generating function nor can it be made so`):
RETURN(FAIL):
fi:

F:=f/mu:


memu1:=simplify(subs(x=1,diff(F,x))):

gu:=[memu1]:

F:=F/x^memu1:

F:=x*diff(F,x):
for i from 2 to N do
 F:=x*diff(F,x):
 gu:=[op(gu),simplify(subs(x=1,F))]:
od:

gu:

end:




#Stat6(n): the expectation, variance, and the 3rd-6th moments
#about the mean of the random variable #occurrences 
#of the pattern 123 in S_n. 
#Also the alpha coefficients up to the sixth
#(for the odd ones you get the square)
#Try: Stat6(n);
Stat6:=proc(n) local gu,lu,i,q,ru,j,x,ku,mu:
 lu:=L20(q):
 gu:=[seq(AveAndMoms(lu[i],q,6),i=1..nops(lu))]:
mu:=[seq(factor(GuessPol([seq(gu[i][j],i=1..nops(gu))],n)),j=1..6)]:
ku:=[op(1..2,mu)]:

for i from 3 to 6 do
if i mod 2=0 then
 ru:=mu[i]/mu[2]^(i/2):
 else
 ru:=mu[i]^2/mu[2]^i:
fi:
 ru:=normal(subs(n=1/x,ru)):
 ru:=taylor(ru,x=0,3):
 ru:=add(coeff(ru,x,j)/n^j,j=0..2):
ku:=[op(ku),ru]:
od:

mu,ku:
end:


#Stat6V(n): Verbose version of Stat6V(n): 
#Try: Stat6V(n);
Stat6V:=proc(n) local gu,mu,i:
gu:=Stat6(n):
mu:=gu[2]:
gu:=gu[1]:

print(`On the Statistics of the Number of Occurrences of the Pattern`):
print(`123 in the Symmetric group`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
print(`Consider the random variable`):
print(`Number of occurrences of the pattern 123 in an n-permutation`):
print(``):
print(`Its expectation is (of course!)`):
print(gu[1]):
print(`and in Maple input form:`):
lprint(gu[1]):
print(``):

print(``):
print(`Its variance is`):
print(gu[2]):
print(`and in Maple input form:`):
lprint(gu[2]):
print(``):

for i from 3 to 6 do
print(`The `, i, `-th moment about the mean is`):
print(gu[i]):
print(`and in Maple input form:`):
lprint(gu[i]):
print(``):
od:

print(`The square of the skewness is, asympotically`):
print(mu[3]):
print(`and in Maple input form:`):
lprint(mu[3]):
print(``):


print(`The kurtosis is, asympotically`):
print(mu[4]):
print(`and in Maple input form:`):
lprint(mu[4]):
print(``):


print(`The square of the fifth alpha coefficient is, asympotically`):
print(mu[5]):
print(`and in Maple input form:`):
lprint(mu[5]):
print(``):


print(`The  sixth alpha coefficient is, asympotically`):
print(mu[6]):
print(`and in Maple input form:`):
lprint(mu[6]):
print(``):

print(`This ends this fascinating article`):

end:






#PCfV(n): Verbose version of PCf(n). Try:
#PCfV(n);
PCfV:=proc(n) local gu,i,lu:
gu:=PCf(n):
lu:=SeqsPC(25,7):
print(`Semi-Rigorously Derived Closed-Form Formulas for the Number of`):
print(`Permutations of size n with Exactly r occurrences of the`):
print(`pattern 123 for r from 0 to 7`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for i from 1 to nops(gu) do
print(`--------------------------------------`):
print(`Theorem Number`, i,`: The number of permutations of length n`):
print(`with  exactly`, i-1, `occurrences of the pattern 123 is `):
print(``):
print(gu[i]):
print(``):
print(`And in Maple input form`):
lprint(gu[i]):
print(``):
print(`The first 25 terms are`):
print(lu[i]):
print(``):
od:

end:

#SeqsPC(N,R): The precomputed Seqs(N,R) for
#N<=25 and R<=7
SeqsPC:=proc(N,R) local gu,i:
option remember:
if N>25 then
print(`N must be <=25, you should use Seqs(N,R)`):
RETURN(FAIL):
fi:

if R>25 then
print(`R must be <=7, you should use Seqs(N,R)`):
RETURN(FAIL):
fi:


gu:=
[[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440,
9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 
91482563640, 343059613650, 1289904147324, 4861946401452], [0, 0, 1, 6, 27, 110,
429, 1638, 6188, 23256, 87210, 326876, 1225785, 4601610, 17298645, 65132550, 
245642760, 927983760, 3511574910, 13309856820, 50528160150, 192113383644, 
731508653106, 2789279908316, 10649977831752], [0, 0, 0, 3, 24, 133, 635, 2807,
11864, 48756, 196707, 783750, 3095708, 12152855, 47500635, 185082495, 719559600
, 2793121080, 10830450780, 41965864794, 162539516448, 629399492330, 
2437072038302, 9437097796918, 36549185005520], [0, 0, 0, 0, 7, 70, 461, 2528, 
12525, 58258, 259787, 1124704, 4765761, 19873150, 81864705, 334052160, 
1353003990, 5447702664, 21830542860, 87145866752, 346793185822, 1376521109436,
5452332482690, 21559064676160, 85125006545645], [0, 0, 0, 1, 9, 74, 507, 3008,
16151, 80889, 385387, 1769705, 7902982, 34539352, 148409950, 629096364, 
2637532977, 10958971820, 45197027220, 185248724362, 755330691518, 3066235194908
, 12400715391073, 49991660982204, 200980915268412], [0, 0, 0, 0, 6, 54, 395, 
2570, 15203, 83382, 431167, 2129734, 10148146, 46993138, 212661944, 944504744,
4130591771, 17833668406, 76170163860, 322377660872, 1353835501123, 
5647648935466, 23424344330803, 96671034640298, 397221475632606], [0, 0, 0, 0, 0
, 37, 387, 2864, 18179, 105082, 568809, 2930602, 14527238, 69819523, 327200581,
1501719377, 6773007550, 30100185693, 132099138291, 573518305776, 2466880451752,
10525239411665, 44590535671158, 187739143883637, 786116211789815], [0, 0, 0, 0,
4, 32, 320, 2544, 17357, 107194, 616449, 3357324, 17513013, 88211386, 431654797
, 2061823804, 9649284793, 44378608890, 201071580333, 899293238364, 
3976973488739, 17414618917290, 75596105867595, 325646307292636, 
1393246785700607]]:

[seq([op(1..N,gu[i])],i=1..R+1)]:

end:











#Wt(pi,q,y): the weight of a permutation pi according to
#123*y[1]#(12,i=1)*y[2]#(12,1=2) ...
#Try
#Wt([1,2,3,4],q,y);
Wt:=proc(pi,q,y) local n,i1,i2,i3,mu:
n:=nops(pi):

mu:=1:

for i1 from 1 to n do
 for i2 from i1+1 to n do
   for i3 from i2+1 to n do
    if pi[i1]<pi[i2] and pi[i2]<pi[i3] then
          mu:=mu*q:
    fi:
   od:
 od:
od:
for i1 from 1 to n do
 for i2 from i1+1 to n do
   if pi[i1]<pi[i2] then
     mu:=mu*y[pi[i1]]:
   fi:
 od:
od:
mu:
end:


#PnDirect(n,q,y):The weight-enumerator of perms of {1, ..., n}
#according to the weight q^(#123)*y[1]#(12,i=1)*y[2]#(12,1=2) ...
#done directly, for checking purposes only! Very slow
PnDirect:=proc(n,q,y) local gu,pi:
gu:=permute(n):
add(Wt(pi,q,y), pi in gu):
end:



#CheckPn(N):Checks Pn(n,q,y) for n from 1 to N 
#Try:
#CheckPn(8);
CheckPn:=proc(N) local n,q,y:
 evalb({seq(PnDirect(n,q,y)-Pn(n,q,y),n=1..N)}={0}):
end: