Brualdi-Goldwasser-Michael problem for maximum permanents of {\rm(0,1)}-matrices

Abstract

Let $\mathscr{U}(n,τ)$ be the set of all {\rm(0,1)}-matrices of order $n$ with exactly $τ$ 0's. Brualdi et al. investigated the maximum permanents of all matrices in $\mathscr{U}(n,τ)$(R.A. Brualdi, J.L. Goldwasser, T.S. Michael, Maximum permanents of matrices of zeros and ones, J. Combin. Theory Ser. A 47 (1988) 207--245.). And they put forward an open problem to characterize the maximum permanents among all matrices in $\mathscr{U}(n,τ)$. In this paper, we focus on the problem. And we characterize the maximum permanents of all matrices in $\mathscr{U}(n,τ)$ when $n^{2}-3n\leqτ\leq n^{2}-2n-1$. Furthermore, we also prove the maximum permanents of all matrices in $\mathscr{U}(n,τ)$ when $σ-kn\equiv0 (mod~k+1)$ and $(k+1)n-σ\equiv0(mod~k)$, where $σ=n^{2}-τ$, $kn\leqσ\leq (k+1)n$ and $k$ is integer.