扭转手镯与转位排序：$S_{16}$ 的转位直径

中文摘要 AI 辅助

转位排序（SBT）旨在寻找将 $n$ 个符号上的排列 $\pi$ 排序为单位排列 $\iota$ 所需的最少转位数。令 $N=n+1$。一个循环目标对 $(\omega,\beta)$ 由一个偶排列 $\omega$ 和一个 $N$-循环 $\beta$ 组成，使得 $\rho=\omega\beta$ 也是一个 $N$-循环。SBT 实例是特殊情况 $(\bar{\iota} {\bar{\pi}}^{-1},\bar{\pi})$，其中 $\bar{\pi}$ 和 $\bar{\iota}$ 分别是与 $\pi$ 和 $\iota$ 对应的 $N$-循环，且 $\bar{\iota} {\bar{\pi}}^{-1}\bar{\pi}=\bar{\iota}$，因此其目标是 $\bar{\iota}$。对于每个指定的无固定点循环类型，并相对于 $\beta$ 指定循环方向，固定内容词编码相应的排列 $\omega$。颜色标识循环，秩记录其方向。当关联乘积 $\rho=\omega\beta$ 是一个 $N$-循环时，词是可实现的。置换等色部分的颜色和移动秩起点给出辅助对称性。结合词旋转和位置反射与秩反转，它们定义了扭转二面体作用。其轨道是扭转手镯，可实现轨道与循环目标对的扩展环面等价类一一对应。这里扩展环面等价是指 Eriksson 等人的环面等价加上反射。这一对应关系产生精确的轨道计数，并直接生成每个可实现扩展环面类的一个代表。转位直径 $TD(n)$ 是 $S_n$ 中最大的转位距离。此前已知 $9\leq TD(16)\leq10$，且 $16$ 是 $n\leq17$ 中唯一未解决的值。结合不动点收缩和结构归约，并对剩余的扭转手镯进行穷举验证，我们证明 $TD(16)=9$，填补了二十五年的空白。

英文摘要

Sorting By Transpositions (SBT) seeks the minimum number of transpositions required to sort a permutation $π$ on $n$ symbols into the identity $ι$. Let $N=n+1$. A cyclic-target pair $(ω,β)$ consists of an even permutation $ω$ and an $N$-cycle $β$ for which $ρ=ωβ$ is an $N$-cycle. An SBT instance is the special case $(\barι{\barπ}^{-1},\barπ)$, where $\barπ$ and $\barι$ encode $π$ and $ι$, and $\barι{\barπ}^{-1}\barπ=\barι$. For a prescribed fixed-point-free cycle type, fixed-content words encode $ω$, with colors distinguishing cycles and ranks recording their orientations relative to $β$. A word is realizable exactly when $ρ=ωβ$ is an $N$-cycle. Permutations of equal-part colors and shifts of rank origins form auxiliary symmetries that, together with word rotation and position reflection coupled to rank inversion, define a twisted dihedral action. Its orbits are twisted bracelets, and its realizable orbits correspond bijectively to extended-toric equivalence classes of cyclic-target pairs, where reflection is adjoined to classical toric equivalence. The correspondence yields a Burnside identity, and our method generates one encoding word for each such class. The transposition diameter $TD(n)$ is the largest transposition distance in $S_n$. Combining fixed-point contraction and reductions of the ambient instance space based on cycle structure with exhaustive verification of every remaining twisted bracelet, we prove $TD(16)=9$, closing a twenty-five-year gap. This result also yields $TD(19)=11$ and, for every $n\equiv1\pmod{3}$ with $n\geq16$, $TD(n)\leq\left\lfloor(2n-2)/3\right\rfloor-1$, improving the previous general upper bound by one for these $n$.

发表机构

• Universidade de Brasília（巴西利亚大学）
• Instituto de Computação, Universidade Federal Fluminense（弗拉门戈联邦大学计算研究所）

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