permutation
Sign in to savethumb|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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Article
37 sectionsContents
- 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
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.
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.