Permutation sign - Knowlpedia

Let $\sigma$ be a finite permutation of $[n]$ . An inversion is a pair of indices $1\le i<j\le n$ for which $\sigma(i)>\sigma(j)$ . Write $\operatorname{inv}(\sigma)$ for the number of these pairs. The sign of $\sigma$ is

\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}\in\{+1,-1\}.

Parity

Thus a permutation is even when its inversion number is even and odd otherwise. The empty permutation has sign +1.

Reference

Konarovskyi, Mathematics 2, §6.1.