Definition:Sign of Permutation on n Letters

Definition
Let $n \in \N$ be a natural number.

Let $N^*_{\le n}$ denote the set of natural numbers $\left\{ {1, 2, \ldots, n}\right\}$.

Let $S_n$ denote the set of all permutations on $N^*_{\le n}$.

Let $\pi \in S_n$ be a permutation of $N^*_{\le n}$.

Let $K$ be the cycle decomposition of $\pi$.

Let each cycle of $K$ be factored into transpositions.

Let $k$ be the total number of transpositions that compose $K$.

The sign of $\pi$ is defined as:
 * $\forall \pi \in S_n: \operatorname{sgn} \left({\pi}\right) = \begin{cases}

1 & : k \text{ even} \\ -1 & : k \text{ odd} \\ \end{cases}$

Also see

 * Sign of Permutation on n Letters is Well-Defined