$S_n$ 的最少操作 Genlex 格雷码：分类与对称性
Genlex Gray codes for $S_n$ with the fewest operations: classification and symmetry
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中文总结 AI 辅助
本文分类了 $S_n$ 中使用最少操作的 genlex 格雷码，证明其构成超阶乘族且具有循环对称性，并在煎饼图中退化为经典 Zaks 序。
中文摘要 AI 辅助
$S_n$ 的格雷码称为 \u201cgenlex\u201d，当共享后缀的单词是连续的。我们确定了使用最少可能操作的 $S_n$ 的 genlex 格雷码：Zaks 递归推广到此类码的一个超阶乘族，且该族之外的任何码都无法达到最小值。该族中的每个码都闭合为循环，且该循环在左平移群下不变，根据操作上的显式准则，该群为 $n$ 阶循环群或 $2n$ 阶二面体群。在煎饼图中，该族简化为单一成员，即经典的 Zaks 序；关于 $W(A_{n-1})$ 的标准抛物链，该序达到总 Coxeter 长度的极端。
英文摘要
A Gray code for $S_n$ is \emph{genlex} when words sharing a suffix are consecutive. We determine the genlex Gray codes for $S_n$ that use the fewest possible operations: Zaks' recursion generalises to a superfactorial family of such codes, and no code outside this family attains the minimum. Every code in the family closes into a cycle, and the cycle is invariant under a group of left translations, cyclic of order $n$ or dihedral of order $2n$ according to an explicit criterion on the operations. In the pancake graph the family reduces to a single member, the classical Zaks order; with respect to the standard parabolic chain of $W(A_{n-1})$, that order attains one extreme of total Coxeter length.
