Stirling numbers of the second kind

Definition

Let $n$ and $k$ be natural numbers. The Stirling number of the second kind $S(n, k)$ is the number of partitions of an $n$-element set into $k$ nonempty blocks. A partition into blocks is a set of pairwise disjoint nonempty subsets whose union is the whole set.

Remarks

Other notations are $\left\{ {n \atop k} \right\}$ and $S_n^{(k)}$. Boundary values: $S(0, 0) = 1$, $S(n, 0) = 0$ for $n \ge 1$, and $S(n, k) = 0$ for $k > n$. Example: $S(4, 2) = 7$. The partitions of $\{1, 2, 3, 4\}$ into two blocks are $1|234$, $2|134$, $3|124$, $4|123$, $12|34$, $13|24$ and $14|23$. The values satisfy a recurrence, and the sum $\sum_k S(n, k)$ is the Bell number $B_n$.

Explanations

L1 human

$S(n, k)$ counts the ways to split $n$ labelled objects into $k$ groups, when the groups have no labels and none is empty. For example, there are $S(3, 2) = 3$ ways to split three people into two teams.