The permanent of an m x n matrix A = (aij), m ≤ n, is defined by

where the summation is over all one-to-one functions σ from {1, … , m} to { 1, …, n}. In other words, the permanent of A is the sum of all the diagonal products of A, that is, all the products of m entries of A no two of which lie in the same row or in the same column. Thus the permanent of A may be evaluated by first multiplying all the row sums of A and then subtracting from the product all terms that contain as factors two or more entries from the same column of A. This is the idea behind the formulas of Binet and of Ryser.