DEF-PAR. Parity of Permutation.
Let $\sigma\in\SS_n$ be a permutation ($n\in\N^+$). The parity of $\sigma$ is
\[\par\sigma = (-1)^k,\]where
\[k = \big\lvert\{(i,j)\in\llbra n\rrbra\times\llbra n\rrbra : i<j\,\land\,\sigma(i)>\sigma(j)\}\big\rvert\]is the number of inversions in $\sigma$. $\sigma$ is called an even permutation if $k$ is even and an odd permutation if $k$ is odd.
If $n\geq 2$ and $\sigma\in\SS_n$,
\[\par\sigma = \prod_{1\leq i<j\leq n}\frac{\sigma(i)-\sigma(j)}{i-j}.\]
For any $n\in\N^+$, $\par:\SS_n\to\{-1,1\}$ is a group homomorphism. We have
\[\ker\par=\AA_n,\]the alternating group of degree $n$.