Definition:Sign of Permutation

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

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

Also denoted as
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 signum.

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

Also see

 * Equivalence of Definitions of Sign of Permutation