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 ^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.