Permutation

Definition

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

The permutations of $X$ form a group 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 of an $n$-element set is $n!$. A derangement 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 $231$.