Proof of the Alon-Tarsi Conjecture for $n=2^rp$

Abstract

The Alon-Tarsi conjecture states that for even $n$, the number of even latin squares of order $n$ differs from the number of odd latin squares of order $n$. Zappa found a generalization of this conjecture which makes sense for odd orders. In this note we prove this extended Alon-Tarsi conjecture for prime orders $p$. By results of Drisko and Zappa, this implies that both conjectures are true for any $n$ of the form $2^rp$ with $p$ prime.