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{\bf D.D. Olesky, Bryan Shader and P. van den Driessche}
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{\bf Permanents of Hessenberg (0,1)-matrices}
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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.

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