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 LE
be, respectively, the number of Latin squares of order n with sign +1 and −1. The Alon-Tarsi conjecture asserts that LE
when n is even. Drisko showed that LE
3) 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 if t ≤ n, then LE
) 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 Lo/LE
, and propose a generalization of the Alon-Tarsi conjecture.