I'm not sure what distribution is desired. Here is one that chooses uniformly among all distinction permutations of partitions of all integers m <= n into nonnegative parts no greater than k. This is done by transposing the Young tableaux for partitions of 2n into at most k + 1 parts, and subtracting 1 to get the parts to be between 0 and k. We then have to select those whose sums are at most n.
ClearAll[getparts, weights];mem : getparts[n_, k_] := mem = getpartsC[ PadRight[IntegerPartitions[2 n, k + 1], {Automatic, k + 1}], n];getpartsC = Compile[{{partitions, _Integer, 2}, {n, _Integer}}, Select[ Function[p, Total@Transpose[UnitStep[p - #] & /@ Range[n]] - 1] /@ partitions, Last[#] >= 0 &] ];mem : weights[n_, k_] := mem = Multinomial @@@ (Tally /@ getparts[n, k])[[All, All, 2]];randsamp = Function[{n, howmany}, Ordering /@ RandomReal[1, {howmany, n}]];ClearAll[nparts];nparts[n_, k_, nsamp_] := With[{parts = getparts[n, k]}, With[{p = RandomChoice[weights[n, k] -> parts, nsamp]}, Compile[{{pp, _Integer, 2}, {samp, _Integer, 2}}, MapThread[#1[[#2]] &, {pp, samp}]][p, randsamp[n, nsamp]] ]];A one-time cost: computing the basic partitions:
getparts[32, 20] // Dimensions // AbsoluteTiming(* {32.0588, {43202, 32}} *)Generating samples is fairly quick (it seems to be an order of magnitude faster than the Combinatorica` code of Kellen Meyers, although there is a claim in a comment, without supporting code, that suggests it might be about the same). A million samples should take a total of about six minutes.
Table[nparts[32, 20, 32], {10^3}] // Dimensions // AbsoluteTiming(* {0.35359, {1000, 32, 32}} *)Getting them in chunks is faster, if you have the memory. A thousand 32x32 arrays is not much, but a million takes more memory than I have RAM, and therefore would be much slower to do all at once.
ArrayReshape[ nparts[32, 20, 32*10^3], {10^3, 32, 32}] // Dimensions // AbsoluteTiming(* {0.070147, {1000, 32, 32}} *)Comparison of this approach and the Combinatorica` approach (MakeArray):
10^3 AbsoluteTiming[Do[MakeArray[], {10^3}]]10^3 AbsoluteTiming[Do[nparts[32, 20, 32], {10^3}]]10^3 AbsoluteTiming[ArrayReshape[nparts[32, 20, 32*10^3], {10^3, 32, 32}];](* {3146.06, 1000 Null} {357.585, 1000 Null} {78.877, 1000 Null}*)Note that you need to add 30 sec. to the nparts results for the one-time computation of the basic partitions.