How not to prove the Alon-Tarsi conjecture

Source abstract

Abstract 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+1 ≢ L 0 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.