Set partition

A set partition is a collection of nonempty, pairwise disjoint subsets called blocks whose union is the original set.

The Stirling number of the second kind $\left\{\begin{array}{c}n\\ k\end{array}\right\}$ counts the set partitions of an $n$-element set into $k$ blocks. It satisfies

The graphical Stirling number ${\left\{\begin{array}{c}n\\ k\end{array}\right\}}_G$ counts partitions of the vertices of a graph $G$ into $k$ nonempty independent sets. Equivalently, it counts proper colourings with $k$ unlabeled nonempty colour classes.