Definition:Sign of Permutation on n Letters

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

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

Let $S_n$ denote the set of 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:

$\forall \pi \in S_n: \sgn \paren \pi = \begin{cases} 1 & : k \text{ even} \\ -1 & : k \text{ odd} \\ \end{cases}$

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