Enumeration of Isomorphism Classes of Self-Orthogonal Latin Squares

Abstract

The numbers of distinct self-orthogonal Latin squares (SOLS) and idempotent SOLS have been enumerated for orders up to and including $9$. The isomorphism classes of idempotent SOLS have also been enumerated for these orders. However, the enumeration of the isomorphism classes of non-idempotent SOLS is still an open problem. By utilising the automorphism groups of class representatives from the already enumerated isomorphism classes of idempotent SOLS, we enumerate the isomorphism classes of non-idempotent SOLS implicitly (i.e. without generating them). New symmetry classes of SOLS are also introduced, based on the number of allowable transformations that may be applied to a SOLS without destroying the property of self-orthogonality, and these classes are also enumerated.