Definition:Sign of Permutation

Definition
Let $S_n$ denote the symmetric group on $n$ letters.

Let $\left \langle {x_k} \right \rangle_{k \in \N^*_n}$ be a sequence in $\R$.

Let $\pi \in S_n$.

Let $\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$ be the product of differences of $\left({x_1, x_2, \ldots, x_n}\right)$.

Let $\pi \cdot \Delta_n \left({x_1, x_2, \ldots, x_n}\right)$ be defined as:
 * $\pi \cdot \Delta_n \left({x_1, x_2, \ldots, x_n}\right) = \Delta_n \left({\pi\left({x_1}\right), \pi\left({x_2}\right), \ldots, \pi\left({x_n}\right)}\right)$

The sign of the permutation $\pi \in S_n$ is defined as:


 * $\operatorname{sgn} \left({\pi}\right) = \dfrac {\Delta_n} {\pi \cdot \Delta_n}$