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{\bf B.~D.~McKay and I.~M.~Wanless}
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{\bf Maximising the Permanent of (0,1)-Matrices and the Number of Extensions of Latin Rectangles}
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Let $k\geq2$, $m\geq5$ and $n=mk$ be integers.  By finding bounds for
certain rook polynomials, we identify the $k\times n$ Latin rectangles
with the most extensions to $(k+1)\times n$ Latin rectangles. 
Equivalently, we find the $(n-k)$-regular subgraphs of $K_{n,n}$ which
have the greatest number of perfect matchings, and the $(0,1)$-matrices
with exactly $k$ zeroes in every row and column which maximise the
permanent.  Without the restriction on $n$ being a multiple of $k$ we
solve the above problem (and the corresponding minimisation problem) for
$k=2$.  We also provide some computational results for small values of
$n$ and $k$. 

Our results partially settle two open problems of Minc and conjectures
by Merriell, and Godsil and McKay.



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