# Definition:Stirling Numbers

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## 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 $1730$.