permutationer

Permutationer is the plural form of the mathematical object known as a permutation in several Scandinavian languages, notably Danish, Norwegian, and Swedish. In English and many other languages, the term permutations is used. A permutation on a finite set X is a bijective function from X to itself, equivalently a rearrangement of the elements of X. If the size of X is n, there are n! different permutationer.

They can be represented in different notations. In one-line notation, a permutation on X = {1,2,3,4} might be written as (2,1,4,3), meaning 1 maps to 2, 2 maps to 1, 3 maps to 4, and 4 maps to 3. Cycle notation expresses the same permutation as disjoint cycles, for example (1 2)(3 4). The set of all permutationer on a finite X forms the symmetric group S_n under composition, with order n!.

Permutationer are central in several areas of mathematics and its applications. They underpin sorting algorithms, counting problems in combinatorics, and various algebraic structures. They also appear in cryptographic contexts and in the study of symmetry in objects and problems.

See also: symmetric group, factorial, cycle notation.