Help26:=proc(): print(` BTseq(N), BTseqE(N), CycDec(pi), WtPi(pi,x), CIP(G,x), Mul(pi,sig), GenGp(S), CubeGp() `):end:

with(combinat):

read `C3.txt`:

BTseq:=proc(N) local x,f,i:
f:=x:
for i from 1 to N+1 do
f:=expand(x+f^2):
f:=taylor(f,x=0,N+2):
f:=add(coeff(f,x,i)*x^i,i=1..N):
od:

[seq(coeff(f,x,i),i=1..N)]:

end:

BTseqE:=proc(N) local x,f,i,i1:
f:=x:
for i from 1 to N+1 do
f:=expand(x+(f^2+subs(x=x^2,f))/2):
f:=taylor(f,x=0,N+2):
f:=add(coeff(f,x,i1)*x^i1,i1=1..N):
od:

[seq(coeff(f,x,i1),i1=1..N)]:

end:

#WtPi(pi,x): The weight of pi
WtPi:=proc(pi,x) local L,i:
L:=CycDec(pi):
mul(x[nops(L[i])],i=1..nops(L)):
end:



#CIP(G,x): The cycle index polynomial of G
CIP:=proc(G,x) local g:
add(WtPi(g,x),g in G):
end:



Mul:=proc(pi,sig) local i:
[seq(sig[pi[i]],i=1..nops(pi))]:
end:


#GenGp(S): The group generated by S
GenGp:=proc(S) local G1,G2,s,g1,g2:
G1:=S:
G2:=G1 union {seq(seq(Mul(s,g1),g1 in G1),s in S)}:
while G1<>G2 do
G1:=G2:
G2:=G2 union {seq(seq(Mul(s,g2),g2 in G2),s in S)}:
od:
G2:
end:


#CubeGp(): The cube symmetry group w/o rotations
CubeGp:=proc() local X,Y,Z:
#[U,L,R,R,B,D]=[1,2,3,4,5,6]
X:=[5,2,1,4,6,3]:
Y:=[5,2,1,4,6,3]:
Z:=[1,5,2,3,4,6]:
GenGp({X,Y,Z}):
end: