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

Let $\mathcal{U}(n,\tau )$ be the set of all {\rm(0,1)}-matrices of order $n$ with exactly $\tau$ 0’s. Brualdi et al. investigated the maximum permanents of all matrices in $\mathcal{U}(n,\tau )$(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 $\mathcal{U}(n,\tau )$. In this paper, we focus on the problem. And we characterize the maximum permanents of all matrices in $\mathcal{U}(n,\tau )$ when $n^2-3n\le \tau \le n^2-2n-1$. Furthermore, we also prove the maximum permanents of all matrices in $\mathcal{U}(n,\tau )$ when $\sigma -kn\equiv 0(modk+1)$ and $(k+1)n-\sigma \equiv 0(modk)$, where $\sigma =n^2-\tau$, $kn\le \sigma \le (k+1)n$ and $k$ is integer.