Odd and Even Permutations

Extract

This note gives a proof for the familiar elementary theorem that if a permutation of the integers 1, …, n (with n ≥ 2) is expressed as a product π₁ of N₁ transpositions and also as a product π₂ of N₂ transpositions, then N₁ and N₂ are both even or both odd (equivalently: N₁ + N₂ is even).

Here, a transposition means an interchange of two of the integers. If the interchange is between adjacent integers it is called an adjacent-trans position.