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There exists a bijection between the set of Latin squares of order $n$ andthe 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 obtaina 1-1 correspondence between the new set of feasible solutions and the set ofLatin 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 toobtain the cardinal of both sets.

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