Definition:Sign of Permutation

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

Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.

Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers.

Let $\pi$ be a permutation of $\N_n$.

Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$.

Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as:


 * $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$

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


 * $\map \sgn \pi = \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 $\map \epsilon \pi$ for $\map \sgn \pi$.

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

1 & : \text {if $\tuple {i, j, k}$ is an even permutation of $\tuple {1, 2, 3}$} \\ -1 & : \text {if $\tuple {i, j, k}$ is an odd permutation of $\tuple {1, 2, 3}$} \\ 0 & : \text {if any two of $\set {i, j, k}$ 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.