Inversions, inversion number, the sign $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$, and even and odd permutations

Let $n\in\mathbb N$ and $\sigma\in S_n=\operatorname{Sym}(n)$. An inversion of $\sigma$ is a pair $(i,j)$ with $i<j<n$ and $\sigma(i)>\sigma(j)$. The inversion set and inversion number are

$\operatorname{Inv}(\sigma):=\{(i,j)\in n\times n:i<j\text{ and }\sigma(i)>\sigma(j)\},\qquad \operatorname{inv}(\sigma):=|\operatorname{Inv}(\sigma)|.$

The sign of $\sigma$ is the integer

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

The permutation is even when its sign is $+1$, equivalently when its inversion number is even, and odd when its sign is $-1$, equivalently when its inversion number is odd. For $n=0$ or $n=1$, every inversion set is empty, so the unique permutation is even.