# Definition:Stirling Numbers of the Second Kind/Definition 1

## Definition

Stirling numbers of the second kind are defined recursively by:

$\displaystyle {n \brace k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ & \\ \displaystyle {n - 1 \brace k - 1} + k {n - 1 \brace k} & : \text{otherwise} \\ \end{cases}$

where:

$\delta_{n k}$ is the Kronecker delta
$n$ and $k$ are non-negative integers.

## Notation

The notation $\displaystyle {n \brace k}$ for Stirling numbers of the second kind is that proposed by Jovan Karamata and publicised by Donald E. Knuth.

Other notations exist.

Usage is inconsistent in the literature.

## Source of Name

This entry was named for James Stirling.

## Technical Note

The $\LaTeX$ code for $\displaystyle {n \brace k}$ is \displaystyle {n \brace k} .

The braces around the n \brace k are important.

The \displaystyle is needed to create the symbol in its proper house display style.