The set of autotopisms of partial Latin squares

Abstract

Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism $\Theta$, the cardinality of the set ${\mathcal{P}\mathcal{LS}}_{\Theta }$ of partial Latin squares which are invariant under $\Theta$ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties studied. It is also seen that the cycle structure of $\Theta$ determines the possible sizes of the elements of ${\mathcal{P}\mathcal{LS}}_{\Theta }$ and the number of those partial Latin squares of this set with a given size. Finally, it is generalized the traditional notion of partial Latin square completable to a Latin square.

20 pages, 4 tables

References in corpus (2)

• Designs based on the cycle structure of a Latin square autotopism
• The 3-dimensional planar assignment problem and the number of Latin squares related to an autotopism

Cited by in corpus (6)

• Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method
• Autoparatopisms of Quasigroups and Latin Squares
• A computational algebraic geometry approach to classify partial Latin rectangles
• Computation of isotopisms of algebras over finite fields by means of graph invariants
• Colouring games based on autotopisms of Latin hyper-rectangles
• Computing autotopism groups of partial Latin rectangles: a pilot study