There are asymptotically the same number of Latin squares of each parity

TL;DR

This paper proves that as the order of Latin squares increases, the number of reduced Latin squares of each parity becomes asymptotically equal, confirming a longstanding conjecture.

Contribution

It confirms the conjecture that the counts of reduced Latin squares of each parity are asymptotically equal for large orders.

Findings

Asymptotic equality of Latin square parities confirmed

Supports conjecture by Stones and Wanless

Provides insight into the structure of Latin squares

Abstract

A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order $n$ there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow\infty$.