Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). One might define an anagram of a word as a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics.

The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×⋯×2×1, which number is called "n factorial" and written "n!".

Permutations occur, in more or less prominent ways, in almost every domain of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science.

In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself (i.e., a map S → S for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a symmetric group. The key to its structure is the possibility to compose permutations: performing two given rearrangements in succession defines a third rearrangement, the composition. Permutations may act on composite objects by rearranging their components, or by certain replacements (substitutions) of symbols.

In elementary combinatorics, the name "permutations and combinations" often refers to two related problems, both counting possibilities to select k distinct elements from a set of n elements, where for k-permutations the order of selection is taken into account, but for k-combinations it is ignored. However k-permutations (also known as partial permutations) do not correspond to permutations as discussed in this article (unless k = n).