Definition:Sign of Permutation

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 $\left({x_1, x_2, \ldots, x_n}\right)$ be an ordered $n$-tuple of real numbers.

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

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:
 * $\pi \cdot \Delta_n \left({x_1, x_2, \ldots, x_n}\right) = \Delta_n \left({x_{\pi \left({1}\right)}, x_{\pi \left({2}\right)}, \ldots, x_{\pi \left({n}\right)}}\right)$

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


 * $\operatorname{sgn} \left({\pi}\right) = \begin{cases}

\dfrac {\Delta_n} {\pi \cdot \Delta_n} & : \Delta_n \ne 0 \\ 0 & : \Delta_n = 0 \end{cases}$

Also denoted as
Some sources use $\epsilon \left({\pi}\right)$ for $\operatorname{sgn} \left({\pi}\right)$.

In physics and applied mathematics, the symbol $e_{ijk}$ can often be found for this concept, referred to as the alternating symbol, defined as:
 * $e_{ijk} = \begin{cases}

1 & : \text{if $\left({i, j, k}\right)$ is an even permutation of $\left({1, 2, 3}\right)$} \\ -1 & : \text{if $\left({i, j, k}\right)$ is an odd permutation of $\left({1, 2, 3}\right)$}\\ 0 & : \text{if any two of $\left\{{i, j, k}\right\}$ are equal}\end{cases}$

Also known as
The sign of a permutation is also known as its signature or signum.

However, on signum is not recommended, in order to keep this concept separate from the signum function on a set of numbers.