# Definition:Parity (Permutation)

## Definition

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

Let $\pi \in S_n$, that is, let $\pi$ be a permutation of $S_n$.

The parity of $\pi$ is defined as:

Parity of $\pi = \begin{cases} \mathrm {Even} & : \map \sgn \pi = 1 \\ \mathrm {Odd} & : \map \sgn \pi = -1 \end{cases}$

where $\map \sgn \pi$ denotes the sign of $\pi$.

## Also defined as

Some sources define the parity of a permutation as $\mathsf{Pr} \infty \mathsf{fWiki}$ defines its sign: that is, as $1$ and $-1$.