Definition:Stirling Numbers

Definition

Stirling numbers come in various forms.

In the below, $n$ and $k$ are always non-negative integers.

Stirling Numbers of the First Kind

Unsigned Stirling Numbers of the First Kind

These are defined recursively by:

$\displaystyle \left[{n \atop k}\right] = \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ \left[{n-1 \atop k-1}\right] + \left({n-1}\right) \left[{n-1 \atop k}\right] & : \text{otherwise} \\ \end{cases}$

where $\delta_{nk}$ is the Kronecker delta.

Signed Stirling Numbers of the First Kind

These are defined recursively by:

$s(n,k) = \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ s(n-1,k-1) - \left({n-1}\right) s(n-1,k) & : \text{otherwise} \\ \end{cases}$

where $\delta_{nk}$ is the Kronecker delta.

Stirling Numbers of the Second Kind

Stirling Numbers of the Second Kind are defined recursively by:

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

Karamata Notation

The notation $\displaystyle \left[{n \atop k}\right]$ and $\displaystyle \left\{{n \atop k}\right\}$ is known as Karamata notation, after Jovan Karamata.

Source of Name

This entry was named for James Stirling.