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

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 $Θ$, 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 $Θ$ 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.

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