# Definition:Stirling Numbers

## Definition

Stirling numbers come in various forms.

In the below:

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

### Unsigned Stirling Numbers of the First Kind

Unsigned Stirling numbers of the first kind are defined recursively by:

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

### Signed Stirling Numbers of the First Kind

Signed Stirling numbers of the first kind are defined recursively by:

$\map s {n, k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ \map s {n - 1, k - 1} - \paren {n - 1} \map s {n - 1, k} & : \text{otherwise} \\ \end{cases}$

### Stirling Numbers of the Second Kind

Stirling numbers of the second kind are defined recursively by:

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

## Karamata Notation

The notation $\displaystyle {n \brack k}$ and $\displaystyle {n \brace k}$ for Stirling numbers is known as Karamata notation.

## Also see

• Results about Stirling numbers can be found here.

## Source of Name

This entry was named for James Stirling.

## Historical Note

This formula for the Stirling numbers of the second kind:

$\displaystyle x^n = \sum_k {n \brace k} x^{\underline k}$

was the reason James Stirling started his studies of the Stirling numbers in the first place.

They were studied in detail in his Methodus Differentialis of $1730$.