Enumeration of Latin squares with conjugate symmetry
A Latin square has six conjugate Latin squares obtained by
uniformly permuting its (row, column, symbol) triples. We say that a Latin
square has conjugate symmetry if at least two of its six conjugates are equal.
We enumerate Latin squares with conjugate symmetry and classify them according
to several common notions of equivalence. We also do similar enumerations
under additional hypotheses, such as assuming the Latin square is
reduced, diagonal, idempotent or unipotent.
Our data corrected an error in earlier literature and suggested
several patterns that we then found proofs for, including
(1) The
number of isomorphism classes of semisymmetric idempotent Latin
squares of order n equals the number of isomorphism classes of
semisymmetric unipotent Latin squares of order n+1, and
(2) Suppose
A and B are totally symmetric Latin squares of order
n≠0 mod 3. If A and B are paratopic then A and B are isomorphic.