---
tag: 0Z22VQH17V
kind: definition
name: Stirling numbers of the first kind
msc: [11B73]
defines: 0KY7GS695G
uses: [0BQMWCNS8S]
sources:
  - url: https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
    locator: Definition by permutation
---

## Definition

Let $n$ and $k$ be natural numbers.
The unsigned Stirling number of the first kind $c(n, k)$ is the number of [[permutations|0BQMWCNS8S]] of an $n$-element set that have exactly $k$ disjoint cycles.
Fixed points count as cycles of length $1$.

## Remarks

Other notations are $\left[ {n \atop k} \right]$ and $|s(n, k)|$. The signed numbers are $s(n, k) = (-1)^{n-k} c(n, k)$.
Example: $c(3, 2) = 3$. The permutations of $\{1, 2, 3\}$ with two cycles are the three transpositions.
The numbers satisfy $\sum_k c(n, k) = n!$ and $x (x + 1) \cdots (x + n - 1) = \sum_k c(n, k)\, x^k$.
The values satisfy a [[recurrence|0C5BFX94J9]] in $n$ and $k$.

## Explanations

### L1 human

Write a permutation as a set of cycles, for example $(1\,2)(3)$.
$c(n, k)$ counts the permutations of $n$ elements that split into exactly $k$ cycles.