Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective mappings 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 σ 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 or signature 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.