Definition:Stirling Numbers of the Second Kind
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Definition
Stirling Numbers of the Second Kind are defined recursively by:
- $\displaystyle \left\{{n \atop k}\right\} = \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ \left\{{n-1 \atop k-1}\right\} + k \left\{{n-1 \atop k}\right\} & : \text{otherwise} \\ \end{cases}$
where:
- $\delta_{nk}$ is the Kronecker delta;
- $n$ and $k$ are always non-negative integers.
Also see
Compare with
Notation
The notation given here is that proposed by Jovan Karamata and publicised by Donald E. Knuth.
Other notations exist, but usage is inconsistent in the literature.
Source of Name
This entry was named for James Stirling.
