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March 2012 How not to prove the Alon-Tarsi conjecture
Douglas S. Stones, Ian M. Wanless
Nagoya Math. J. 205: 1-24 (March 2012). DOI: 10.1215/00277630-1543769


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 LnE and LnO be, respectively, the number of Latin squares of order n with sign +1 and 1. The Alon-Tarsi conjecture asserts that LnELnO when n is even. Drisko showed that Lp+1ELp+1O(modp3) for prime p3 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 tn, then Ln+1ELn+1O(modt3) 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 n9, discuss asymptotics for LO/LE, and propose a generalization of the Alon-Tarsi conjecture.


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Douglas S. Stones. Ian M. Wanless. "How not to prove the Alon-Tarsi conjecture." Nagoya Math. J. 205 1 - 24, March 2012.


Published: March 2012
First available in Project Euclid: 1 March 2012

zbMATH: 1245.05011
MathSciNet: MR2891163
Digital Object Identifier: 10.1215/00277630-1543769

Primary: 05B15
Secondary: 11B50

Rights: Copyright © 2012 Editorial Board, Nagoya Mathematical Journal


Vol.205 • March 2012
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