How not to prove the Alon-Tarsi Conjecture

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 Lne ≠ Lno when n is even. Drisko showed that Lp+1e ≠ Lp+1o 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 Ln+1e ≠ Ln+1o mod t3 for t ≤ n only if t=n and n is prime, thereby showing that Drisko's methodology 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 generalisation of the Alon-Tarsi Conjecture.

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