Enumeration of Latin squares with conjugate symmetry. Issue 2 (16th November 2021)

• Record Type: Journal Article
• Authors: McKay, Brendan D.; Wanless, Ian M.
• Is Part Of: Journal of combinatorial designs. Volume 30:Issue 2(2022)
• Page Start: 105
• Page End: 130
• Publication Date: 2021-11-16
• Subjects: idempotent -- Latin square -- semisymmetric -- symmetric -- totally symmetric -- unipotent; Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21814
• Languages: English
• ISSNs: 1063-8539

Abstract

Abstract: 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.