Signature (permutation)

In mathematics, when X is a finite set of 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 ${\displaystyle \sigma }$ 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 ${\displaystyle x<y}$ and ${\displaystyle \sigma (x)>\sigma (y)}$.

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