I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related problem?

Here is the problem:

Let $A$ and $B$ be disjoint $k$-subsets of $\mathbb{Z}/n\mathbb{Z}$. Consider $S(A,B)=\sum_{x\in A}x - \sum_{y\in B}y$. As $(A,B)$ ranges over all possible ordered pairs of disjoint $k$-subsets of $\mathbb{Z}/n\mathbb{Z}$, how are the sums $S(A,B)$ distributed over the elements of $\mathbb{Z}/n\mathbb{Z}$? More precisely, for how many of the $\binom{n}{k}\binom{n-k}{k}$ choices of $(A,B)$ is $S(A,B)$ equal to each of the elements of $\mathbb{Z}/n\mathbb{Z}$?

Again I am just looking for references. I actually have a solution in the case that n is prime, but I assume the result is known for more general n. I would be interested in any leads.