Permutation

Definition

Permutation

A permutation of a set is a bijective function from to itself:

Thus every element of has exactly one image and exactly one preimage. A permutation may move elements, but it neither duplicates nor removes them.

Finite Sets

If , then has permutations, where is the factorial. After ordering the elements of as , every permutation corresponds to the ordered arrangement

Example

A three-cycle

On , define

This is a permutation because every element appears exactly once as an output. In cycle notation, it is written .