Permutaatiom

Permutaatiom is a term used in some contexts to denote the act or result of permuting elements. In mathematics, the standard term for this concept is permutation, which refers to a rearrangement of the elements of a set. The word appears in various languages as a variant of permutaatio or permute, but in formal work the precise term is typically permutation.

In a finite set of n distinct elements, a permutation is a bijection from the set to itself, equivalent to an ordering of the elements. The collection of all such permutations forms the symmetric group S_n under composition. The identity permutation leaves the order unchanged, and every permutation can be decomposed into disjoint cycles. The number of permutations of n distinct elements is n!, and permutations of multisets (with repeated elements) are counted by multinomial coefficients, taking into account repeated items.

Common examples help illustrate the idea. For the set {A, B, C}, the six permutations are ABC, ACB, BAC, BCA, CAB, and CBA. In algorithmic contexts, permutations are generated by methods such as lexicographic generation or Heap’s algorithm. Key concepts include parity (whether a permutation is even or odd) and transpositions, which swap two elements and generate all permutations.

Applications of permutation concepts are widespread, including sorting, shuffling, scheduling, cryptography, and design of experiments. In genetics or molecular biology, related rearrangements of DNA segments are sometimes described using permutation-like ideas, reflecting changes in gene order.

See also: permutation, permutation group, factorial, cycle notation, combination.