On the maximum of the permanent of (I − A)

Abstract Let ω n s and ω ˜ n s denote the convex sets of n × n doubly substochastic matrices and row substochastic matrices with the sum of all entries equal to s , respectively. In this paper, we give the upper bound of the permanent of I − A for A ∈ ω ˜ n s . We also give the upper bound of the permanent of I − A for A ∈ ω n s , where either n is even, or n is odd and s ≤ n − 1 . For the case when n is odd and n − 1 s ≤ n , we give some discussion and conjectures.