Stirling number

There are two kinds of Stirling numbers: Stirling numbers of the first kind and Stirling numbers of the second kind. They appear in many situations in combinatorics.

Stirling Numbers of the First Kind

The Stirling number of the first kind, , is the number of permutations of an -element set with exactly cycles.

For example, because (writing all our permutations in cycle notation) we have the permutations .

Stirling Numbers of the Second Kind

The Stirling number of the second kind, , is the number of partitions of an -element set into exactly subsets.

For example, because we have the partitions .