permutation

Also known as arrangement, rearrangement, shuffle

thumb|120 px|According to the first meaning of permutation, each of the six rows is a different permutation of three distinct balls|alt=The six different possible ways to order three balls of different colors: (red, green, blue), (red, blue, green), (green, red, blue), (green, blue, red), (blue, red, green), and (blue, green, red). In mathematics, a permutation of a set can mean one of two different things: an arrangement of its members in a sequence or linear order, or the act or process of changing the linear order of an ordered set.

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A permutation is an arrangement of items in a specific sequence or order—for example, the different ways you can line up three colored balls. Permutations matter in mathematics because they help us count and analyze all the possible ways to arrange or rearrange a set of things, which has applications in probability, logic, and many other fields.

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Commons category: Permutations

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Article

37 sections

Contents

History
Definition
Notations
Two-line notation
One-line notation
Cycle notation
Canonical cycle notation
Composition of permutations
Other uses of the term ''permutation''
''k''-permutations of ''n''
Permutations with repetition
Permutations of multisets
Circular permutations
Properties
Cycle type
Conjugating permutations
Order of a permutation
Parity of a permutation
Matrix representation
Permutations of totally ordered sets
Ascents, descents, runs, exceedances, records
Foata's transition lemma
Inversions
Permutations in computing
Numbering permutations
Algorithms to generate permutations
Random generation of permutations
Generation in lexicographic order
Generation with minimal changes
Generation of permutations in nested swap steps
Applications
See also
Notes
References
Bibliography
Further reading
External links

An example of the first meaning is the six permutations (orderings) of the set {1, 2, 3}: written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory.