Definition:Stirling Numbers of the Second Kind/Definition 2

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Definition

Stirling numbers of the second kind are defined as the coefficients $\displaystyle {n \brace k}$ which satisfy the equation:

$\displaystyle x^n = \sum_k {n \brace k} x^{\underline k}$

where $x^{\underline k}$ denotes the $k$th falling factorial of $x$.


Notation

The notation $\displaystyle {n \brace k}$ 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.


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 $1730$.


Technical Note

The $\LaTeX$ code for \(\displaystyle {n \brace k}\) is \displaystyle {n \brace k} .

The braces around the n \brace k are important.

The \displaystyle is needed to create the symbol in its proper house display style.


Sources