Parity of a permutation

The parity of a permutation tells us whether a permutation can be made by swapping pairs of elements an even or odd number of times. If it can be done with an even number of swaps, the permutation is called even; if an odd number, it is called odd.

Important properties

• Every permutation is either even or odd, never both.

• The identity permutation (which leaves everything in place) is even.

• The parity of a permutation does not change if you write it in a different way using swaps.

• The product (composition) of two permutations: even × even = even, odd × odd = even, even × odd = odd.