Generating a list of integers that sums to zero

Question

Given a pair of integers n and k, I want to generate all lists of integers of length n, containing integers between -k and +k, such that the total of the list is equal to zero. How can I do this in the most performant way?

Naively, I could do something like

Select[Tuples[Range[-k,k],n],Total[#]==0&]

but this means I'm generating far more Tuples than I need, and for relatively small values of n and k, this fails with error SystemException[MemoryAllocationFailure]

I do not need all permutations. Is there a less wasteful way to generate all such lists?

Answer 1

f[n_, k_] := IntegerPartitions[0, {n}, Range[-k, k]]

f[4, 2]
(*    {{2, 2, -2, -2}, {2, 1, -1, -2}, {2, 0, 0, -2}, {2, 0, -1, -1},
       {1, 1, 0, -2}, {1, 1, -1, -1}, {1, 0, 0, -1}, {0, 0, 0, 0}}    *)

The fourth argument of IntegerPartitions allows you to get only a reduced number of solutions instead of all solutions.

If you need all permutations of these:

g[n_, k_] := Join @@ Permutations /@ f[n, k]

g[4, 2]
(*    {{2, 2, -2, -2}, {2, -2, 2, -2}, {2, -2, -2, 2},
       ...
       {-1, 1, 0, 0}, {-1, 0, 1, 0}, {-1, 0, 0, 1}, {0, 0, 0, 0}}    *)