## Abstract

Let k ≥ 2, m ≥ 5 and n = mk be integers. By finding bounds for certain rook polynomials, we identify the k × n Latin rectangles with the most extensions to (k+1) × 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.

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 5 |

Issue number | 1 |

Publication status | Published - 1 Dec 1998 |

Externally published | Yes |