Brualdi-Goldwasser-Michael problem for maximum permanents of {\rm(0,1)}-matrices
Tingzeng Wu, Xiangshuai Dong, Huazhong Lü
TL;DR
The paper resolves the Brualdi-Goldwasser-Michael problem for maximum permanents of $n\times n$ (0,1)-matrices with a fixed number of zeros by deriving a closed-form upper bound $\\mu(n,\\tau)$ and characterizing extremal configurations. Using a factorial-product maximization argument, it proves that the maximum is achieved by block-structured matrices $K_{n,\\sigma}$ when $\sigma-kn\\equiv 0 \pmod{k+1}$ and $(k+1)n-\\sigma\equiv 0 \pmod{k}$, with the explicit value $\\mu(n,\\tau)=((k+1)!)^{(\\sigma-kn)/(k+1)}(k!)^{((k+1)n-\\sigma)/k}$. For the range $2n+1 \le\\sigma \le 3n$ (equivalently $n^{2}-3n \le\\tau \le n^{2}-2n-1$) a detailed inductive classification identifies the extremal matrices and provides the exact permanents, involving specific families of block-structured matrices determined by $\\sigma-2n$ mod 3 and parity of $3n-\\sigma$. Maple-assisted verification anchors the base case $n=8$, after which the induction yields a complete description in this regime. The results illuminate the interplay between combinatorial matrix structure and permanent optimization and advance the understanding of permanents near the high-density region of ones.
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.