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How not to prove the Alon-Tarsi conjecture

Published online by Cambridge University Press:  11 January 2016

Douglas S. Stones  and
Ian M. Wanless
Douglas S. Stones
Affiliation:
School of Mathematical Sciences and Clayton School of Information Technology Monash University, Victoria 3800, Australiathe_empty_element@yahoo.com
Ian M. Wanless
Affiliation:
School of Mathematical Sciences Monash University, Victoria 3800, Australiaian.wanless@monash.edu
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Abstract

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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 LEn and Lon be, respectively, the number of Latin squares of order n with sign +1 and −1. The Alon-Tarsi conjecture asserts that LEn ≠ Lon when n is even. Drisko showed that LEp+1 ≢ Lop+1 (mod p3) for prime p ≥ 3 and asked if similar congruences hold for orders of the form pk + 1, p + 3, or pq + 1. In this article we show that if t ≤ n, then LEn+1 ≢ L0n+1 (mod t3) 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 Lo/LE, and propose a generalization of the Alon-Tarsi conjecture.

Keywords

05B1511B50
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 205 , March 2012 , pp. 1 - 24
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

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