On the number of even latin squares of even order
Carolin Hannusch
TL;DR
The paper investigates the Alon-Tarsi conjecture on the enumeration of even versus odd latin squares for even order and introduces a parity-switching map $m_{i,j}$ that preserves latin structure while potentially flipping parity under certain cycle conditions. It proves that for any reduced odd latin square of even order there exist indices $i$ and $j$ such that $m_{i,j}(\lambda)$ is a reduced even latin square, establishing $Erls(n)\geq Orls(n)$ and linking this to the conjectured inequality $Els(n)\geq Ols(n)$ if the conjecture holds. The construction relies on analyzing the cycle structure of row permutations $\sigma_i$ and $\sigma_j$ and their product, with a key claim that $m_{i,j}$ is nontrivial whenever $\sigma_i\sigma_j$ is not an $n$-cycle. An explicit example and a discussion of parity behavior for even latin squares illustrate the method and its limitations, highlighting how cycle interactions influence parity outcomes and the potential for progress toward the Alon-Tarsi conjecture.
Abstract
We recall the Alon-Tarsi conjecture on the number of even latin squares. We introduce a map which switches the parity of a latin square under certain requirements. An example is included.