Brualdi-Goldwasser-Michael problem for maximum permanents of {\rm(0,1)}-matrices
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arXiv
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| Format: | Preprint |
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2025
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| author | Wu, Tingzeng Dong, Xiangshuai Lü, Huazhong |
| author_facet | Wu, Tingzeng Dong, Xiangshuai Lü, Huazhong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Brualdi-Goldwasser-Michael problem for maximum permanents of {\rm(0,1)}-matrices Wu, Tingzeng Dong, Xiangshuai Lü, Huazhong Combinatorics 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. |
| title | Brualdi-Goldwasser-Michael problem for maximum permanents of {\rm(0,1)}-matrices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.12787 |