---
tag: 0BQMWCNS8S
kind: definition
name: Permutation
msc: [05A05]
defines: 05RWFKVTJ7
sources:
  - url: https://en.wikipedia.org/wiki/Permutation
    locator: Definition
---

## Definition

Let $X$ be a set.
A permutation of $X$ is a bijection $\sigma : X \to X$.

The permutations of $X$ form a [[group|G5S1M9]] under composition, with the identity map as identity element.
This group is the symmetric group on $X$, written $\operatorname{Sym}(X)$.
For $X = \{1, \dots, n\}$ it is written $S_n$.

## Remarks

For a finite set $X = \{1, \dots, n\}$, a permutation is often written in one-line notation as the sequence $\sigma(1) \sigma(2) \cdots \sigma(n)$. In this form it is an arrangement of the elements in a row.
The [[number of permutations|0CEBCS0E0E]] of an $n$-element set is $n!$.
A [[derangement|0TH61JR04G]] is a permutation with no fixed point.

## Explanations

### L1 human

A permutation is a rearrangement of the elements of a set.
For $\{1, 2, 3\}$, the map that sends $1 \mapsto 2$, $2 \mapsto 3$ and $3 \mapsto 1$ is a permutation. In one-line notation it is $231$.