Difference between revisions of "Permutations"
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==Origin== | ==Origin== | ||
[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'' | ||
| − | *[ | + | *[https://en.wikipedia.org/wiki/14th_century 14th Century] |
==Definitions== | ==Definitions== | ||
*1: often major or fundamental [[change]] (as in [[character]] or condition) based primarily on rearrangement of existent elements <the system has gone through several permutations>; also : a [[form]] or variety resulting from such [[change]] <[[technology]] available in various permutations> | *1: often major or fundamental [[change]] (as in [[character]] or condition) based primarily on rearrangement of existent elements <the system has gone through several permutations>; also : a [[form]] or variety resulting from such [[change]] <[[technology]] available in various permutations> | ||
*2a : the [[act]] or [[process]] of changing the lineal order of an ordered set of objects | *2a : the [[act]] or [[process]] of changing the lineal order of an ordered set of objects | ||
==Description== | ==Description== | ||
| − | 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 [ | + | 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]. |
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!". | 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!". | ||
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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 [ | + | 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 "[ | + | 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] |
[[Category: Mathematics]] | [[Category: Mathematics]] | ||
Latest revision as of 02:23, 13 December 2020
Origin
Middle English permutacioun exchange, transformation, from Anglo-French, from Latin permutation-, permutatio, from permutare
Definitions
- 1: often major or fundamental change (as in character or condition) based primarily on rearrangement of existent elements <the system has gone through several permutations>; also : a form or variety resulting from such change <technology available in various permutations>
- 2a : the act or process of changing the lineal order of an ordered set of objects
Description
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" 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).[1]
