A Stirling number of the second kind, $S (r,k)$, is defined to be the number of ways one can partition an $r$-element set into $k$ subsets.
Consider the following problem:
You have $r$ distinguishable balls and $n$ undistinguishable boxes. If you are allowed to place balls into these boxes with no restrictions, how many different ways of doing so are there?
The anwser is $\sum_{k=1}^{n} S (r,k)$.
You can nicely use this model for many combinatorial problems.
I would like to ask if anyone has seen, or knows of a problem where one would sum over $r$ instead, i.e. where something like $\sum_{k=i}^{j} S (k,n)$ would appear.
This just a general interest question; it came up in a combinatorial methods class. Many thanks for any anwsers/comments.