Symmetries of partial Latin squares

Abstract

In this paper, we study symmetries (autoparatopisms) of partial Latin squares. Let s(n) be the minimum number of non-empty cells in a partial Latin square of order n with a trivial autoparatopism group. We show 15(6n−7)≤s(n)≤12(3n−3) for all n≥5. We also show that, if G is a finite group, then there exists a partial Latin square whose autoparatopism group is isomorphic to G (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.