Parity of a permutation

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 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 Sn. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (εσ), 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. Consider the permutation σ of the set defined by and In one-line notation, this permutation is denoted 34521. It can be obtained from the identity permutation 12345 by three transpositions: first exchange the numbers 2 and 4, then exchange 3 and 5, and finally exchange 1 and 3. This shows that the given permutation σ is odd. Following the method of the cycle notation article, this could be written, composing from right to left, as There are many other ways of writing σ as a composition of transpositions, for instance σ = (1 5)(3 4)(2 4)(1 2)(2 3), but it is impossible to write it as a product of an even number of transpositions. The identity permutation is an even permutation.