Definition:Stirling Numbers of the Second Kind/Definition 1

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

• Equivalence of Definitions of Stirling Numbers of the Second Kind

• Definition:Stirling's Triangles

• Definition:Stirling Numbers of the First Kind

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: $(46)$