Some properties of the permanent of (1,-1)-matrices

Let ω n denote the set of all n × n −(1, −1)-matrices. In [5] E. I. II. Wang posed the following problem. Is there a decent upper bound for [per A] when A ∊ ω n is nonsingular? In this paper we conjecture that the best possible bound is the permanent of a matrix in ω n which contains exactly n 1 negative entries all occurring in its main diagonal. This conjecture is affirmed by the study of a large class of matrices in Ω n . Moreover, some other interesting results concerning the permanent function in Ω n are given.