Definition:Stirling Numbers of the Second Kind/Definition 1
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Definition
Stirling numbers of the second kind are defined recursively by:
- $\ds {n \brace k} := \begin{cases}
\delta_{n k} & : k = 0 \text{ or } n = 0 \\ & \\ \ds {n - 1 \brace k - 1} + k {n - 1 \brace k} & : \text{otherwise} \\ \end{cases}$
where:
- $\delta_{n k}$ is the Kronecker delta
- $n$ and $k$ are non-negative integers.
Notation
The notation $\ds {n \brace k}$ 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.
Also see
Source of Name
This entry was named for James Stirling.
Technical Note
The $\LaTeX$ code for \(\ds {n \brace k}\) is \ds {n \brace k}
.
The braces around the n \brace k
are important.
The \ds
is needed to create the symbol in its proper house display style.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(46)$