Category:Definitions/Sign of Permutation

This category contains definitions related to Sign of Permutation.
Related results can be found in Category:Sign of Permutation.

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}$