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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.

Maximising the Permanent of (0,1)-Matrices and the Number of Extensions of Latin Rectangles