Let $r\in\mathbb{N}$, and let $\sigma\in S_r$, where $S_r$ is the set from \reftext{def:permutation-initial-segment-2026a}{the permutation definition}. An inversion of $\sigma$ is a pair $(i,j)$ such that $1\le i<j\le r$ and $\sigma(i)>\sigma(j)$. Let $N(\sigma)$ denote the number of inversions of $\sigma$. The sign of $\sigma$ is the number

defined by

if $N(\sigma)$ is \reftext{def:even-odd-natural-numbers-2026a}{even}, and by

if $N(\sigma)$ is \reftext{def:even-odd-natural-numbers-2026a}{odd}.