Permanents of Doubly Substochastic Matrices

Let ωn,k denote the convex polytope of doubly substochastic matrices with sub-defect k. Let h(A) and l(A) denote the maximum and minimum diagonals of A ϵ ωn,k respectively. In this paper, we study the relation between the permanent function and the maximum (minimum) diagonals, which are defined below. More specifically, we give the upper bound of the permanent function on ωn,k in terms of the h-function. We also study the upper bound of the permanent function on both the product and direct product of matrices in ωn,k