# 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 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 L_{n+1}^{e} ≠
L_{n+1}^{o} mod t^{3} 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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