with(combinat):

reverse := proc(L::list) return [seq(L[nops(L)-i+1], i = 1 .. nops(L))] end proc:

mypartition := proc(d ::integer) 
local X;
X := reverse(partition(d)):
return [seq(reverse(X[i]),i=1..numbpart(d))];
end proc:

myconjpart := proc (L::list) local preK, K; 
description "I needed to have a conjugate partition function that is in decreasing order."; 
return reverse(conjpart(reverse(L)));
end proc;

smash := proc (u, v) 
description "smash two vectors together"; 
[seq(u[i], i = 1 .. nops(u)), seq(v[i], i = 1 .. nops(v))] 
end proc;

dimModule := proc (LL::list, k) local L, m; 
 description "this just applies a formula from Fulton and Harris pg 77";

 L := smash(LL, [seq(0, i = 1 .. k)]); m := `*`(seq(seq((L[i]-L[j]+j-i)/(j-i), j = i+1 .. k), i = 1 .. k-1)); 
 if k < nops(LL) then m := 0 end if; 
 return m 
end proc;

numbclass:= proc (L::list) local m, f, g, r, top, i, j, c, LL; 

description "see exercise in http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf for the formula for computing the number of elements in a given conjugacy class.  This procedure computes the number of elements in the conjugacy class corresponding to a given partition as imput"; 

top := max(seq(L[k], k = 1 .. nops(L))); 
for i to top do c := 0; 
	for j to nops(L) do 
		if L[j] = i then c := c+1 end if 
	end do; 
	r[i] := c 
end do;
LL := [seq(r[i], i = 1 .. top)];
m := `+`(seq(i*LL[i], i = 1 .. nops(LL))); 
f := `*`(seq(i^LL[i], i = 1 .. nops(LL))); 
g := `*`(seq(factorial(LL[i]), i = 1 .. nops(LL))); 
return factorial(m)/(f*g) 
end proc;


mults := proc (degree::integer, numfacts::integer, maxdim::integer)
local M, Par, i, p, X, IP2, N, NN, k, numpar, temp, stopper, d, n, count, myindicator, myoutput, breaker, cc, K; 
description "This procedure computes the multiplicites of the irreducible modules in the decomposition of S^d(A_1\otimes \dots \otimes A_n).  It expects three integers, the degree - d, the number of factors -n, and the maximum dimension of the A_i - maxdim."; 

d := degree; n := numfacts; 
if d = 1 then return start*over end if;

 K := partition(d);
 
 N := [seq(numbclass(K[i]), i = 1 .. nops(K))];
 
 M := character(d);
 Par := mypartition(d);
 numpar := nops(Par);
 count := 1;
 for i to numpar do 
 	if evalb(nops(Par[i]) <= maxdim) then myindicator[count] := i; count := count+1 end if 
end do; 

stopper := count-1;
count := 'count';

for i to stopper do
	X[myindicator[i]] := S[Par[myindicator[i]]] 
end do;
i := 'i'; cc := 1;

if n <= 10 then breaker := [seq(1, i = 1 .. n), seq(0, i = n+1 .. 11)]; 
	for i[1] to stopper do
		 for i[2] from i[1] to i[1]+breaker[2]*(stopper-i[1]) do
		 	 for i[3] from i[2] to i[2]+breaker[3]*(stopper-i[2]) do
			 	for i[4] from i[3] to i[3]+breaker[4]*(stopper-i[3]) do
					for i[5] from i[4] to i[4]+breaker[5]*(stopper-i[4]) do
						for i[6] from i[5] to i[5]+breaker[6]*(stopper-i[5]) do
							for i[7] from i[6] to i[6]+breaker[7]*(stopper-i[6]) do
								for i[8] from i[7] to i[7]+breaker[8]*(stopper-i[7]) do
									for i[9] from i[8] to i[8]+breaker[9]*(stopper-i[8]) do
										for i[10] from i[9] to i[9]+breaker[10]*(stopper-i[9]) do
										temp := simplify((`+`(seq(N[q]*(`*`(seq(M[myindicator[i[p]], q], p = 1 .. n))), q = 1 .. nops(Par))))/factorial(d));
										 if 0 < temp then myoutput[cc] := [temp, [seq(X[myindicator[i[p]]], p = 1 .. n)]]; cc := cc+1 end if 
										end do
									end do
								end do
							end do
						end do
					end do
				end do
			end do
		 end do
	end do
end if; 
									
return [seq(myoutput[i], i = 1 .. cc-1)]
end proc;


prettymults := proc (n, d, max) 
local p, i, box, pp; 
for p to nops(mults(n, d, max)) do
	pp := nops(op(2, op(p, mults(n, d, max)))); 
	 for i to pp do 
 		box[2*i-1] := op(i, op(2, op(p, mults(n, d, max)))); box[2*i] := `&bigotimes` 
	end do; 
print(Matrix([seq(box[i], i = 1 .. 2*pp-1)]), `&oplus`, op(1, op(p, mults(n, d, max)))) 
end do

 end proc;


##examples ##

mults(3,6,3);

dimModule([3,2,1],4);