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.