Help:=proc() : print(`G(m,n,q) , RatM(M,N,q), G1(m,n,q) `): end:


#[a[1],a[2], ..., a[M]]
#N>=a[1]>=...>=a[M]>=0
#G(M,N,q):The generating polynomial whose coeff. of q^n
#is the number of ways of arranging
#n chocolate pieces that fit into an M by N box
#(left-justified and top-justified)
G:=proc(M,N,q)
option remember:

if M=0 or N=0 then
  1:


elif M<0 or N<0 then
0:
else

#Case 1: top-right piece has been eaten
sort(expand(G(M,N-1,q)+
#case 2: top-right has not (yet) been eaten
q^N*G(M-1,N,q))):

fi:

end:


RatM:=proc(M,N,q): normal(G(M,N,q)/G(M-1,N,q)):end:

RatMN:=proc(M,N,q): normal(RatM(N,N,q)/RatM(N-1,M,q)):end:


#G(M,2)=(1-q^(M+2))*(1-q^(M+1))/(1-q)/(1-q^2):
#G(M,3)=(1-q^(M+3))*(1-q^(M+2))*(1-q^(M+1))/(1-q)/(1-q^2)/(1-q^3)


G1:=proc(M,N,q) local i:

normal(mul((1-q^(M+i))/(1-q^i),i=1..N)):


end:




#N>=a1 >= a2>=  ... >= aM>=0
#v=    v=            v=
#N>=b1>= b2>= ... >=   bM>=0
#
#PP2(M,N,q): the gen. poly for 2 by M by N (two-layered)
#choc. box
#
PP2:=proc(M,N,q):
expand(add(add( PP2a(a1,b1),a1=0..N),b1=0..a1)):
end: