# 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.

## 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

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(45)$