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 (3 PAPn). In this paper, we prove that, given a Latin square isotopism Θ, we can add some linear con-straints to the 3 PAPn in order to obtain a 1−1 correspondence between the new set offeasible solutions and the set of Latin squares of orden having Θ in their autotopism group. Moreover, we use Gröbner bases in order to describe an algorithm that allows one to obtain the cardinal of both sets.