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According to the first meaning of permutation, each of the six rows is a different permutation of three distinct balls. 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. [1]
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...
A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
A cyclic permutation consisting of a single 8-cycle. There is not widespread consensus about the precise definition of a cyclic permutation. Some authors define a permutation σ of a set X to be cyclic if "successive application would take each object of the permuted set successively through the positions of all the other objects", [1] or, equivalently, if its representation in cycle notation ...
Multiplying a matrix M by either or on either the left or the right will permute either the rows or columns of M by either π or π −1.The details are a bit tricky. To begin with, when we permute the entries of a vector (, …,) by some permutation π, we move the entry of the input vector into the () slot of the output vector.
Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ , or less commonly the Latin lower case e .
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Considering the symmetric group S n of all permutations of the set {1, ..., n}, we can conclude that the map sgn: S n → {−1, 1} that assigns to every permutation its signature is a group homomorphism. [2] Furthermore, we see that the even permutations form a subgroup of S n. [1] This is the alternating group on n letters, denoted by A n. [3]