Compilation of IntegerPartitions

Questions
1

I need to compile a function to render it as fast as possible because it is used recurrently. Inside this function I use IntegerPartitions. However, it seems impossible to correctly compile code using IntegerPartitions. For instance, for the following test code

F = Compile[{{k, _Integer}, {d, _Integer}, {s, _Integer}}, Block[{ls},
   ls = IntegerPartitions[s, {k}, Range[d]];
   Return[ls]],
  CompilationTarget -> "C"]

the answer is “Compiled expression {{3,1,1},{2,2,1}} should be a rank 1 tensor of machine-size integers” when F[3,7,5] is executed, whether the CompilationTarget is C or WVM. However, with ls={{3,1,1},{2,2,1}} it is executed flawlessly. Has anyone an idea what is the reason for this oddity?

2

I know nothing about compiling, but the following

F = Compile[{{k, _Integer}, {d, _Integer}, {s, _Integer}}, 
  IntegerPartitions[s, {k}, Range[d]]]
With[{s = 8, k = 4, d = 5}, IntegerPartitions[s, {k}, Range[d]]]
With[{s = 8, k = 4, d = 5}, F[k, d, s]]

gives the message “Could not complete external evaluation at instruction 9; proceeding with uncompiled evaluation”. Maybe IntegerPartitions cannot be compiled.

What is the purpose of Block in your code?

3

I get the same message. Well, Block has no purpose in this test code. I would use Block together with other instructions that would run faster if I could compile them. The function I want to compile is

SP[k_, d_, s_] := Block[{sum, ls, ln, zeros, part, t, numP},
   sum = 0.;
   Do[
    ls = IntegerPartitions[s, {i}, Range[d]]; ln = Length[ls];
    Do[
     part = Join[ConstantArray[0, k - Length[ls[[j]]]], ls[[j]]];
     t = Tally[part];
     numP = k!/Product[t[[l, 2]]!, {l, 1, Length[t]}];
     sum = sum + numP*Apply[Times, Map[y, part]],
     {j, 1, ln}],
    {i, 1, k}];
   Return[sum]];

Map[y, part] has still to be replaced with Table[y[[i + 1]], {i, part}], otherwise it cannot be compiled. This code calculates the sum of all y[s1] * y[s2] * … * y[sk] where the si satisfy s1 +…+ sk = s and 0 <= si <= d. The y are a binomial distribution. Have you an idea how this can be compiled?