On the number of Latin squares

TL;DR

This paper advances the understanding of Latin squares by enumerating Latin rectangles, analyzing divisibility properties, deriving formulas involving permanents, exploring extremal combinatorial values, and studying symmetry group proportions.

Contribution

It provides new enumeration formulas, divisibility results, and asymptotic symmetry properties for Latin squares of order up to 11, addressing several open questions.

Findings

Number of Latin rectangles with 11 columns and various rows determined.

Number of reduced Latin squares divisible by a factorial related to order.

Formula for Latin squares using permanents of signed matrices.

Abstract

We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order $n$ is divisible by $f!$ where $f$ is a particular integer close to $\frac12n$, (3) provide a formula for the number of Latin squares in terms of permanents of $(+1,-1)$-matrices, (4) find the extremal values for the number of 1-factorisations of $k$-regular bipartite graphs on $2n$ vertices whenever $1\leq k\leq n\leq11$, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases.