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 L_n^e and L_n^o be respectively the number of Latin squares of order n with sign +1 and -1. The Alon-Tarsi Conjecture asserts that L_n^e ≠ L_n^o when n is even. Drisko showed that L_p+1^e ≠ L_p+1^o mod p³ 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 L_n+1^e ≠ L_n+1^o 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 L^o/L^e and propose a generalisation of the Alon-Tarsi Conjecture.

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