Symmetries of partial Latin squares

In this paper, we study symmetries (autoparatopisms) of partial Latin squares. Let s(n)s(n) be the minimum number of non-empty cells in a partial Latin square of order nn with a trivial autoparatopism group. We show 15(6n−7)≤s(n)≤12(3n−3) for all n≥5n≥5. We also show that, if GG is a finite group, then there exists a partial Latin square whose autoparatopism group is isomorphic to GG (as are its autotopism and automorphism groups). Computational methods are also introduced, and are used to study symmetries of partial Latin squares of small orders; the source code has been made available as supplementary material.