Consider the problem of partitioning a set with $n$ elements into at most $k$ parts. Clearly, when $k\geq n$, the answer is $B_{n}$ (the $n$th Bell number).
What happens when $k<n$ (beyond that this is a partial sum of Stirling numbers of the second kind)? Do these numbers have a name (something like incomplete or partial Bell numbers)? Is there a good reference to this topic?
Edit: As discussed in the comments, this is the sequence A102661 in the The On-Line Encyclopedia of Integer Sequences.