# Definition:Stirling Numbers of the Second Kind/Definition 2

## 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.

## 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.