## Abstract

The sign of a Latin square is -1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. Let L ^{E} _{n} and L ^{o} _{n} be, respectively, the number of Latin squares of order n with sign +1 and -1. The Alon-Tarsi conjecture asserts that L ^{E} _{n} ≠ L ^{o} _{n} when n is even. Drisko showed that L ^{E} _{p+1} ≡ L ^{o} _{p+1} (mod p ^{3}) for prime p ≥ 3 and asked if similar congruences hold for orders of the form p ^{k} + 1, p + 3, or pq + 1. In this article we show that if t ≤ n, then L ^{E} _{n+} ≡ L ^{o} _{n+1} (mod t ^{3}) only if t = n and n is an odd prime, thereby showing that Drisko's method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to n ≤ 9, discuss asymptotics for L ^{o}/L ^{E}, and propose a generalization of the Alon-Tarsi conjecture.

Original language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Nagoya Mathematical Journal |

Volume | 205 |

DOIs | |

Publication status | Published - 2012 |

Externally published | Yes |