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

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.