Definition:Stirling Numbers of the Second Kind

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


 * $\displaystyle \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

 * Definition:Stirling's Triangles

Compare with

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

Notation
The notation given here is that proposed by and publicised by.

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