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

TL;DR

This paper characterizes the maximum permanents of (0,1)-matrices with a fixed number of zeros in specific parameter ranges, extending previous open problems in combinatorial matrix theory.

Contribution

It provides explicit characterizations of maximum permanents for matrices in al{U}(n, au) within certain al{U}(n, au) parameter ranges, solving an open problem.

Findings

Characterized maximum permanents for al{U}(n, au) when n^2 - 3n al{ au} al{n}^2 - 2n - 1.

Derived formulas for maximum permanents under modular conditions involving al{ au} and n.

Extended understanding of maximum permanents in (0,1)-matrices with fixed zero counts.

Abstract

Let $\mathscr{U}(n,\tau)$ 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 $\mathscr{U}(n,\tau)$(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 $\mathscr{U}(n,\tau)$. In this paper, we focus on the problem. And we characterize the maximum permanents of all matrices in $\mathscr{U}(n,\tau)$ when $n^{2}-3n\leq\tau\leq n^{2}-2n-1$. Furthermore, we also prove the maximum permanents of all matrices in $\mathscr{U}(n,\tau)$ when $\sigma-kn\equiv0 (mod~k+1)$ and $(k+1)n-\sigma\equiv0(mod~k)$, where $\sigma=n^{2}-\tau$, $kn\leq\sigma\leq (k+1)n$ and $k$ is integer.