permutations - meaning and definition. What is permutations
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What (who) is permutations - definition

CHANGE OF ORDERING IN A (MATHEMATICAL) SET
Cycle notation; Permute; Arrangement number; Cycle representation; Cyclic notation; NPr; Cycle type; Permutes; Permutations; Cycle decomposition (group theory); Permuter; Next permutation; Permuting; One-line notation; Calculating permutations; K-permutation; Circular notation; Disposition (math); Permutation algorithms; Permutation formula; Two-line notation; Cycle structure; Cycle shape; Rothe diagram
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  • Each of the six rows is a different permutation of three distinct balls
  • Permutations of multisets
  • Composition of permutations corresponding to a multiplication of permutation matrices.

permutation         
¦ noun each of several possible ways in which a set or number of things can be ordered or arranged.
?Mathematics the action of changing the arrangement of a set of items.
?Brit. a selection of a specified number of matches in a football pool.
Derivatives
permutational adjective
Origin
ME (in the sense 'exchange, barter'): via OFr. from L. permutatio(n-), from permutare (see permute).
permute         
¦ verb submit to a process of alteration, rearrangement, or permutation.
Origin
ME: from L. permutare 'change completely'.
Permutation         
·noun Barter; exchange.
II. Permutation ·noun Any one of such possible arrangements.
III. Permutation ·noun The act of permuting; exchange of the thing for another; mutual transference; interchange.
IV. Permutation ·noun The arrangement of any determinate number of things, as units, objects, letters, ·etc., in all possible orders, one after the other;
- called also alternation. ·cf. Combination, ·noun, 4.

Wikipedia

Permutation

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.

Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, 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). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.

Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.

The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.

Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3, 1, 2) mentioned above is described by the function α {\displaystyle \alpha } defined as

α ( 1 ) = 3 , α ( 2 ) = 1 , α ( 3 ) = 2 {\displaystyle \alpha (1)=3,\quad \alpha (2)=1,\quad \alpha (3)=2} .

The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition (performing two given rearrangements in succession), which results in another rearrangement. As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set { 1 , 2 , , n } {\displaystyle \{1,2,\ldots ,n\}} that are considered for studying permutations.

In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.

Examples of use of permutations
1. Other permutations of partnership have also proliferated.
2. When they reported this to him, he approved all permutations.
3. There are all these permutations." Reader comments (0) No comments have so far been submitted.
4. It wasn‘t enough to indulge his fascination for the game‘s mathematics and its endless winning permutations.
5. Long stretches in opposition sometimes drive parties toward defeatism, utopianism, isolationism or permutations of all three.