On the Permanent of a Doubly Stochastic Matrix
Herbert S. Wilf · Canadian Journal of Mathematics · 1966
If is an n X n matrix, the permanent of A, Per A, is defined by 1 where the sum is over all permutations. If A is doubly stochastic (i.e., nonnegative with row and column sums all equal to 1), then it has been conjectured that Per A ⩾ n!/nn. When confronted with a vaguely similar problem about determinants, M. Kac (1) observed that the computation of minima can often be aided by knowledge of various averages. In this spirit we calculate here the average permanent of a class of doubly stochastic matrices and thereby obtain upper bounds for the minima. These turn out to be surprisingly sharp.