Let ω8n and ~ω8n denote the convex sets of 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 ϵ ~ω8n. We also give the upper bound of the permanent of I - A for A ϵ ~ω8n, 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.

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