The 3-dimensional planar assignment problem and the number of Latin squares related to an autotopism

TL;DR

This paper establishes a method to count Latin squares with a specific autotopism by adding linear constraints to the 3-dimensional planar assignment problem and employing Gr"obner bases for computation.

Contribution

It introduces a novel linear constraint approach to link Latin squares with autotopisms to feasible solutions of a constrained 3PAP and uses Gr"obner bases for enumeration.

Findings

Linear constraints create a bijection for Latin squares with autotopisms.

Algorithm using Gr"obner bases computes the number of such Latin squares.

Provides a new algebraic method for counting Latin squares with symmetries.

Abstract

There exists a bijection between the set of Latin squares of order $n$ and the set of feasible solutions of the 3-dimensional planar assignment problem ($3PAP_n$). In this paper, we prove that, given a Latin square isotopism $\Theta$, we can add some linear constraints to the $3PAP_n$ in order to obtain a 1-1 correspondence between the new set of feasible solutions and the set of Latin squares of order $n$ having $\Theta$ in their autotopism group. Moreover, we use Gr\"obner bases in order to describe an algorithm that allows one to obtain the cardinal of both sets.