An odd permutation is a permutation obtainable from an odd number of two-element swaps, i.e., a permutation with permutation symbol equal to . For initial set 1,2,3,4, the twelve odd permutations are those with one swap (1,2,4,3, 1,3,2,4, 1,4,3,2, 2,1,3,4, 3,2,1,4, 4,2,3,1) and those with three swaps (2,3,4,1, 2,4,1,3, 3,1,4,2, 3,4,2,1, 4,1,2,3, 4,3,1,2).

For a set of elements and , there are odd permutations (D'Angelo and West 2000, p. 111), which is the same as the number of even permutations. For , 2, ..., the numbers are given by 0, 1, 3, 12, 60, 360, 2520, 20160, 181440, ... (OEIS A001710).