Even permutation

An even permutation is a permutation that can be produced by an even number of exchanges (called transpositions). For example, (1 3 2)=(1 2)(1 3) is an even permutation. See symmetric group for an elaboration.

An identity permutation is an even permutation as (1)=(1 2)(1 2).

The composition of two even permutations is again an even permutation, and so is the inverse of an even permutation: the even permutations of n letters form a group, the alternating group on n letters, denoted by A_n. This is a subgroup of the symmetric group S_n and contains n/2 permutations.

An odd permutation is a permutation which is not an even permutation, equivalently, it is a product by odd number of transpositions.

See fifteen puzzle for a classic application.