交错群中的循环排序
Circular sorting in the alternating group
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中文总结 AI 辅助
该研究探讨交错群中用3-循环排序偶置换的最大步数，针对不同模4余值的$n$确定或分析了排序次数，证明较大值会无限次出现。
中文摘要 AI 辅助
对称群$S_n$由对换生成，利用对换对排列进行排序的问题已得到充分研究。近期Adin、Alon和Roichman研究了相关问题：对圆上的$n$个点进行排序，并给出所需相邻交换的最大次数公式。这等价于用相邻对换将任意排列转化为循环置换$(1,2,\boldsymbol{\text{…}},n)$的幂所需的相邻对换次数。本工作的重点是交错群$A_n$中的类似问题，$A_n$由3-循环生成，即：使用3-循环而非对换，在交错群中将偶置换转化为$(1,2,\boldsymbol{\text{…}},n)$的幂所需的最大步数是多少？我们针对偶数$n$和$n \boldsymbol{\text{≡}} 1 \boldsymbol{\text{mod}} 4$的情况精确确定了该数值；对于$n \boldsymbol{\text{≡}} 3 \boldsymbol{\text{mod}} 4$的情况，我们证明排序次数可取两个可能值之一，并给出明确构造，证明较大值会无限次出现。
英文摘要
The symmetric group $S_n$ is generated by transpositions, and problems of sorting permutations using transpositions are well studied. In recent work, Adin, Alon, and Roichman studied the related problem of sorting $n$ points on a circle, and gave a formula for the maximum number of adjacent swaps required. This is equivalent to the number of adjacent transpositions required to transform any permutation into a power of the cyclic permutation $(1,2,\ldots, n)$. The focus of this work is an analogous question in the alternating group $A_n$, which is generated by $3$-cycles. That is, using 3-cycles instead of transpositions, what is the maximum number of steps required to transform an even permutation into a power of $(1,2,\ldots, n)$ in the alternating group? We determine this number exactly for even $n$ and $n \equiv 1 \pmod{4}$. For $n \equiv 3 \pmod{4}$, we show that the sorting number can take one of two possible values and give explicit constructions demonstrating that the larger value occurs infinitely often.
