Enumeration of Latin squares with conjugate symmetry

TL;DR

This paper enumerates Latin squares with conjugate symmetry, classifies them under various equivalences, corrects previous errors, and uncovers new patterns linking different classes of Latin squares.

Contribution

It provides the first comprehensive enumeration of conjugate symmetric Latin squares and establishes new relationships between classes like semisymmetric, unipotent, and totally symmetric Latin squares.

Findings

Number of semisymmetric idempotent Latin squares of order n equals that of semisymmetric unipotent Latin squares of order n+1.

Paratopic totally symmetric Latin squares of order n (not divisible by 3) are necessarily isomorphic.

Corrected earlier enumeration errors and identified new structural patterns.

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 $n\not\equiv0\bmod3$. If $A$ and $B$ are paratopic then $A$ and $B$ are isomorphic.