Permanent - Definition

Definition

The permanent of an n-by-n matrix A = (a_i,j) is defined as

The sum here extends over all elements σ of the symmetric group S_n, i.e. over all permutations of the numbers 1, 2, ..., n.

For example (2×2 matrix),

The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account. If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). A formula similar to Laplace's for the development of a determinant along a row or column is also valid for the permanent; all signs have to be ignored for the permanent.

The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument. In his monograph, Minc (1984) uses Per(A) for the permanent of rectangular matrices, and uses per(A) when A is a square matrix. Muir (1882) uses the notation .

The word, permanent originated with Cauchy (1812) as “fonctions symétriques permanentes” for a related type of functions, and was used by Muir (1882) in the modern, more specific, sense.