\begin{definition}[Definition:Sign of Permutation on n Letters]
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:
:$\forall \pi \in S_n: \sgn \paren \pi = \begin{cases}
1 & : k \text{ even} \\
-1 & : k \text{ odd} \\
\end{cases}$
\end{definition}