perms

In mathematics, a permutation, often abbreviated perms, is an arrangement of objects. Formally, a permutation of a set S is a bijection from S to S, equivalently a reordering of S’s elements. If S has n elements, a permutation yields one of the n! possible orders.

For a finite set of size n, there are n! permutations. This factorial growth underpins the structure of the symmetric group S_n, which collects all permutations under composition.

Permutations can be described in different notations. One-line notation lists the image of each element; cycle notation expresses permutations as disjoint cycles such as (1 3 2). Any permutation can be decomposed into transpositions, and its parity is even or odd according to the number of transpositions in such a decomposition.

The symmetric group S_n is a group under composition, with the identity permutation as its neutral element. A derangement is a permutation with no fixed points, a classical counting problem linked to n! via inclusion–exclusion.

Example: on the set {1,2,3}, the six permutations are 123, 132, 213, 231, 312, 321. In cycle notation these correspond to the identity, transpositions, and 3-cycles, illustrating how different representations capture the same rearrangements.

Applications of permutations include combinatorics, probability, sorting algorithms, and cryptography. Permutations contrast with combinations, which count selections without regard to order; permutations focus on the order in which elements appear.