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.

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Tag	`0BQMWCNS8S`
URI	`m2://0BQMWCNS8S`
Kind	definition
Writing	W1 stated
MSC	05A05
Defines	Permutation

Permutation

Definition

Remarks

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