Permutation (disambiguation)

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In mathematics, permutation relates to the act of arranging all the members of a set into some sequence or order.

Permutation Arrangements of a list or set

In mathematics, permutation is the act of arranging the members of a set into a sequence or order, or, if the set is already ordered, rearranging (reordering) its elements—a process called permuting. 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.

Permutation may also refer to:

Iteration is the repetition of a process in order to generate a sequence of outcomes. The sequence will approach some end point or end value. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration.

Permutation (music)

In music, a permutation (order) of a set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. Different permutations may be related by transformation, through the application of zero or more operations, such as transposition, inversion, retrogradation, circular permutation, or multiplicative operations. These may produce reorderings of the members of the set, or may simply map the set onto itself.

<i>Permutation</i> (Amon Tobin album) 1998 studio album by Amon Tobin

Permutation is the third studio album by Brazilian electronic music producer Amon Tobin and the second under his own name. It was released in 1998, just over a year after Bricolage. The album was a success for Tobin and found him playing sold-out shows at the Montreal Jazz Festival, the Knitting Factory in New York and the Coachella Valley Music and Arts Festival. He went on to release Supermodified in 2000.

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Bijection one to one and onto mapping of a set X to a set Y

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.

Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G. The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1,2,...,n} then, Sym(M), the symmetric group on n letters is usually denoted by Sn.

Symmetric group automorphism group of a set; the group of bijections on a set, whose group operation is function composition

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group Sn defined over a finite set of n symbols consists of the permutation operations that can be performed on the n symbols. Since there are n! possible permutation operations that can be performed on a tuple composed of n symbols, it follows that the number of elements of the symmetric group Sn is n!.

Descent may refer to:

Group theory branch of mathematics that studies the algebraic properties of groups

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Proof may refer to:

Duality may refer to:

Difference or differences may refer to:

In mathematics, anticommutativity is a specific property of some non-commutative operations. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments. Swapping the position of two arguments of an antisymmetric operation yields a result, which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation, such as addition.

Cycle or cyclic may refer to:

A sign is an entity which indicates another entity.

In statistics, resampling is any of a variety of methods for doing one of the following:

  1. Estimating the precision of sample statistics by using subsets of available data (jackknifing) or drawing randomly with replacement from a set of data points (bootstrapping)
  2. Exchanging labels on data points when performing significance tests
  3. Validating models by using random subsets

Let be a finite permutation group acting on a set . A sequence

Permutation graph graph whose vertices represent the elements of a permutation

In mathematics, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation if it is prime with respect to the modular decomposition.

Abstract algebra branch of mathematics

In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

Order or ORDER or Orders may refer to:

Jorge Luis Borges and mathematics

Jorge Luis Borges and mathematics concerns several modern mathematical concepts found in certain essays and short stories of Argentinian author Jorge Luis Borges (1899-1986), including concepts such as set theory, recursion, chaos theory, and infinite sequences, although Borges' strongest links to mathematics are through Georg Cantor's theory of infinite sets, outlined in "The Doctrine of Cycles". Some of Borges' most popular works such as "The Library of Babel", "The Garden of Forking Paths", "The Aleph", an allusion to Cantor's use of the Hebrew letter aleph to denote cardinality of transfinite sets, and "The Approach to Al-Mu'tasim" illustrate his use of mathematics.

In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into two packets and then the two packets are interleaved. Beginning with an ordered set, mathematically a riffle shuffle is defined as a permutation on this set containing 1 or 2 rising sequences. The permutations with 1 rising sequence are the identity permutations.