Geodesic
Permanents of Hessenberg (0,1)-matrices
The electronic journal of combinatorics, Tome 12 (2005)
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Let $P\left( m,n\right) $ denote the maximum permanent of an $n$-by-$n$ lower Hessenberg $(0,1) $-matrix with $m$ entries equal to 1. A "staircased" structure for some matrices achieving this maximum is obtained, and recursive formulas for computing $P\left( m,n\right) $ are given. This structure and results about permanents are used to determine the exact values of $P\left( m,n\right) $ for $n \leq m \leq 8n/3$ and for all $\hbox{nnz}(H_n)-\hbox{nnz}(H_{\lfloor n/2 \rfloor}) \leq m \leq \hbox{nnz}(H_n)$, where $\hbox{nnz}(H_n)=(n^2+3n-2)/2$ is the maximum number of ones in an $n$-by-$n$ Hessenberg $(0,1)$-matrix.
DOI : 10.37236/1967
Classification : 15A15, 15B36, 05B20, 05C50
Mots-clés : maximum permanent, binary Hessenberg matrix, staircase 
@article{10_37236_1967,
     author = {D. D. Olesky and Bryan Shader and P. van den Driessche},
     title = {Permanents of {Hessenberg} (0,1)-matrices},
     journal = {The electronic journal of combinatorics},
     year = {2005},
     volume = {12},
     doi = {10.37236/1967},
     zbl = {1086.15007},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1967/}
}
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D. D. Olesky; Bryan Shader; P. van den Driessche. Permanents of Hessenberg (0,1)-matrices. The electronic journal of combinatorics, Tome 12 (2005). doi: 10.37236/1967

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