# 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.