Stirling Partition Number of a Set

Stirling numbers of the second kind count the number of ways of partitioning a set of $n$ elements into $k$ non-empty disjoint subsets whose union is the original set. This is equivalent to counting the number of ways of distributing $n$ distinguishable balls into $k$ indistinguishable boxes. The problem turns out to reduce to counting surjections from one finite set to another, with a twist.