| [https://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English] ''permutacioun'' [[exchange]], [[transformation]], from Anglo-French, from [[Latin]] ''permutation''-, ''permutatio'', from ''permutare'' | | [https://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English] ''permutacioun'' [[exchange]], [[transformation]], from Anglo-French, from [[Latin]] ''permutation''-, ''permutatio'', from ''permutare'' |
− | 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 [http://en.wikipedia.org/wiki/Anagram anagram] of a word as a permutation of its [[letters]]. The [[study]] of permutations in this sense generally belongs to the field of [http://en.wikipedia.org/wiki/Combinatorics combinatorics]. | + | 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 [https://en.wikipedia.org/wiki/Anagram anagram] of a word as a permutation of its [[letters]]. The [[study]] of permutations in this sense generally belongs to the field of [https://en.wikipedia.org/wiki/Combinatorics combinatorics]. |
| 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]]. | | 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 [http://en.wikipedia.org/wiki/Algebra algebra] and particularly in [http://en.wikipedia.org/wiki/Group_theory 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 [http://en.wikipedia.org/wiki/Symmetric_group 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 ([[substitution]]s) of [[symbols]]. | + | In [https://en.wikipedia.org/wiki/Algebra algebra] and particularly in [https://en.wikipedia.org/wiki/Group_theory 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 [https://en.wikipedia.org/wiki/Symmetric_group 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 ([[substitution]]s) of [[symbols]]. |
− | In elementary combinatorics, the name "[http://en.wikipedia.org/wiki/Permutations_and_combinations permutations and combinations]" 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 do not correspond to permutations as discussed in this article (unless k = n).[http://en.wikipedia.org/wiki/Permutation] | + | In elementary combinatorics, the name "[https://en.wikipedia.org/wiki/Permutations_and_combinations permutations and combinations]" 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 do not correspond to permutations as discussed in this article (unless k = n).[https://en.wikipedia.org/wiki/Permutation] |