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In mathematics, when X is a finite set with at least two elements, the permutations of 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 of a permutation σ {\displaystyle \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 > σ.
The sign, signature, or signum of a permutation σ is denoted sgn and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol , 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
where N is the number of inversions in σ.