How not to prove the Alon-Tarsi conjecture

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 $L_n^{\mathrm{E}}$ and $L_n^{\mathrm{O}}$ be, respectively, the number of Latin squares of order $n$ with sign $+1$ and $-1$. The Alon-Tarsi conjecture asserts that $L_n^{\mathrm{E}}\ne L_n^{\mathrm{O}}$ when $n$ is even. Drisko showed that $L_{p+1}^{\mathrm{E}}\not\equiv L_{p+1}^{\mathrm{O}}(mod{p}^3)$ for prime $p\ge 3$ 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 $t\le n$, then $L_{n+1}^{\mathrm{E}}\not\equiv L_{n+1}^{\mathrm{O}}(mod{t}^3)$ 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 $n\le 9$, discuss asymptotics for $L^{\mathrm{O}}/L^{\mathrm{E}}$, and propose a generalization of the Alon-Tarsi conjecture.