Sign of a Permutation

Let $r\in\mathbb{N}$, and let $\sigma\in S_r$, where $S_r$ is the set from 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 even, and by

if $N(\sigma)$ is odd.