Definition:Parity

Integer
Let $$z \in \Z$$ be an integer.

The parity of $$z$$ is whether it is even or odd.

That is:
 * an integer of the form $$z = 2 n$$, where $$n$$ is an integer, is of even parity;
 * an integer of the form $$z = 2 n + 1$$, where $$n$$ is an integer, is of odd parity.

Also see Odd Integer 2n + 1.


 * If $$z_1$$ and $$z_2$$ are either both even or both odd, $$z_1$$ and $$z_2$$ have the same parity.
 * If $$z_1$$ is even and $$z_2$$ is odd, then $$z_1$$ and $$z_2$$ have opposite parity.

Permutation
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 $$\pi = \begin{cases} \mathrm {Even} & : \sgn \left({\pi}\right) = 1 \\ \mathrm {Odd} & : \sgn \left({\pi}\right) = -1 \end{cases}$$

where $$\sgn \left({\pi}\right)$$ is the sign of $\pi$.

Also see

 * Parity Group.