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 $n\not\equiv 0 \mathrm{mod}3$. If $A$ and $B$ are paratopic then $A$ and $B$ are isomorphic.

中文翻译：

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具有共轭对称性的拉丁方的枚举

一个拉丁方阵有六个共轭拉丁方阵，通过对其（行、列、符号）三元组进行统一排列而获得。如果它的六个共轭中至少有两个相等，我们就说拉丁方具有共轭对称性。我们列举了具有共轭对称性的拉丁方阵，并根据几种常见的等价概念对它们进行分类。我们还在其他假设下进行了类似的枚举，例如假设拉丁方是减少的、对角的、幂等的或单能的。我们的数据纠正了早期文献中的错误，并提出了几种模式，然后我们找到了证据，包括（1）半对称幂等拉丁阶次方的同构类的数量$n$ 等于半对称单能拉丁阶次方同构类的数量 $n+1$, 和 (2) 假设 $\mathrm{一种}$ 和 $乙$ 是完全对称的拉丁阶次方 $n\not\equiv 0 \mathrm{模组}3$. 如果$\mathrm{一种}$ 和 $乙$ 那么是副题的 $\mathrm{一种}$ 和 $乙$ 是同构的。