The number S(n,r) of ways of partitioning a set of n elements into r non-empty subsets. For example, the set {1,2,3,4} can be partitioned into two non-empty subsets in the following ways:

So S(4, 2) = 7. Clearly, S(n,1) = 1 and S(n,n) = 1. It can be shown that

Rather like the binomial coefficients, the Stirling numbers occur as coefficients in certain identities. They are named after the Scottish mathematician James Stirling (1692–1770).