Permutations

Permutations

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). For example, an anagram of a word is 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 is commonly denoted as "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 SS 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 k-permutations, or partial permutations, are the sequences 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.

Read more about Permutations:  History, Permutations in Group Theory, Permutations in Combinatorics, See Also

Other articles related to "permutations, permutation":

Permutations - See Also
... Mathematics portal Alternating permutation Binomial coefficient Combination Combinatorics Convolution Cyclic order Cyclic permutation Even and odd permutations ...
Alexander's Star - Permutations
... two positions, giving a theoretical maximum of 30!×230 permutations ... This value is not reached for the following reasons Only even permutations of edges are possible, reducing the possible edge arrangements to 30!/2 ... It would be impossible to swap all 15 pairs (an odd permutation), so a reducing factor of 214 is applied ...
List Of Permutation Topics - Combinatorics of Permutations
... Cycle notation Cycles and fixed points Cyclic order Direct sum of permutations Enumerations of specific permutation classes Factorial Falling factorial Permutation matrix ...
Significance Analysis Of Microarrays - Running SAM
... generate a List of Significant Genes, Delta Table, and Assessment of Sample Sizes Permutations are calculated based on the number of samples Block Permutations Blocks are ... A minimum of 1000 permutations are recommended the number of permutations is set by the user when imputing correct values for the data set to run SAM ...
Multiple Zeta Function - Symmetric Sums in Terms of The Zeta Function - Theorem 2(Hoffman)
... To prove this, note first that the sign of is positive if the permutations of cycle-type are even, and negative if they are odd thus, the left-hand side of ... isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e ...

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