Stirling Numbers

Stirling numbers of the second kind in discrete mathematics are used to show numerous combinatoric properties like partitioning a set, (a number of ways to write an integer of a set), and forming a recurrence relation.

The Stirling Number of a second kind is denoted by \(S\left(n,k\right)\) or \(\binom{n}{k}\) counts the number of ways to partition a set of \(n\) elements into \(k\) non-empty sets. 

For example, \(S\left(3,2\right)\)\(=3\), because there are three ways to partition three objects, namely \(a,b\) and \(c\) into two groups. \(\left(a\right)\left(bc\right),\left(b\right)\left(ac\right)\) and \(\left(c\right)\left(ab\right).\)

Another combinatorial property of stirling numbers of the second kind is that it can be shown in terms of a recurrence relation:

\(S\left(n,k\right)=kS\left(n-1,k\right)+S\left(n-1,k-1\right),1\le k<n\)

Stirling numbers of \(s\left(n,k\right)\)of the first kind relates to the amount or number of partitions of a set \(n\) including \(k\) cycles.