On some questions concerning permanents of (1,−1)-matrices

Abstract

Let Ω n denote the set of alln×n-(1,−1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ω n such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2 k −1,k≧2, even for singularA∈Ω n . Moreover, we prove that perA≠0 ifA∈Ω n ,n=2 k −1, and we derive new results concerning the divisibility of the permanent in Ω n by powers of 2.