# Parity of a permutation

## Property in group theory / From Wikipedia, the free encyclopedia

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In mathematics, when *X* is a finite set with at least two elements, the permutations of *X* (i.e. the bijective functions from *X* to *X*) fall into two classes of equal size: the **even permutations** and the **odd permutations**. If any total ordering of *X* is fixed, the **parity** (**oddness** or **evenness**) of a permutation $\sigma$ of *X* can be defined as the parity of the number of inversions for *σ*, i.e., of pairs of elements *x*, *y* of *X* such that *x* < *y* and *σ*(*x*) > *σ*(*y*).

The **sign**, **signature**, or **signum** of a permutation *σ* is denoted sgn(*σ*) and defined as +1 if *σ* is even and −1 if *σ* is odd. The signature defines the **alternating character** of the symmetric group S_{n}. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (*ε*_{σ}), which is defined for all maps from *X* to *X*, and has value zero for non-bijective maps.

The sign of a permutation can be explicitly expressed as

- sgn(
*σ*) = (−1)^{N(σ)}

where *N*(*σ*) is the number of inversions in *σ*.

Alternatively, the sign of a permutation *σ* can be defined from its decomposition into the product of transpositions as

- sgn(
*σ*) = (−1)^{m}

where *m* is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well-defined.^{[1]}