Upper bounds for permanents of (1, − 1)-matrices

Let Ω _n denote the set of alln×n (1, ? 1)-matrices. In 1974 E. T. H. Wang posed the following problems: Is there a decent upper bound for |perA| whenAσΩ _n is nonsingular? We recently conjectured that the best possible bound is the permanent of the matrix with exactlyn?1 negative entries in the main diagonal, and affirmed that conjecture by the study of a large class of matrices in Ω _n . Here we prove that this conjecture also holds for another large class of (1, ?1)-matrices which are all nonsingular. We also give an upper bound for the permanents of a class of matrices in Ω _n which are not all regular.