Definition:Stirling Numbers of the Second Kind

Definition
Stirling Numbers of the Second Kind are defined recursively by:


 * $$\left\{{n \atop k}\right\} = \begin{cases}

\delta_{n k} & : k = 0 \text{ or } n = 0 \\ \left\{{n-1 \atop k-1}\right\} + k \left\{{n-1 \atop k}\right\} & : \text{otherwise} \\ \end{cases}$$ where:
 * $$\delta_{nk}$$ is the Kronecker delta;
 * $$n$$ and $$k$$ are always non-negative integers.

Also see

 * Stirling's Triangles

Compare with

 * Stirling Numbers of the First Kind
 * Pascal's Triangle

Notation
The notation given here is that proposed by Jovan Karamata and publicised by Donald E. Knuth.

Other notations exist, but usage is inconsistent in the literature.