PAR#DEF. 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.
For any $\sigma\in\SS_n$ ($n\geq 2$),
\[\par\sigma = \prod_{1\leq i<j\leq n}\frac{\sigma(i)-\sigma(j)}{i-j}.\]
For any $n\in\N^+$,
\[\par : \SS_n \to \{-1,1\}\]is a group homomorphism and we have
\[\ker\par = \AA_n,\]the alternating group of degree $n$.