arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1907.09451math.CO
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模式回避置换幂

Pattern-Avoiding Permutation Powers

Amanda Burcroff, Colin Defant

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16 pages, 1 figure
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摘要：

最近，Bóna和Smith定义了强模式回避，称如果$π$和$π^2$都回避$τ$，则置换$π$强烈地回避模式$τ$。他们证明，对于每个正整数$k$，$S_{k^3}$中存在一个置换，它强烈地避免了$123\cdots (k+1)$。我们利用Robinson-Schensted-Knuth对应解决了这个猜想，证明了这种排列的个数至少为$k^{k^3/2+O(k^3/\log k)}$，至多为$k^{2k^3+O(k^3/\log k)}$ .我们列举了$3$阶的$231$ -避免置换，并进一步给出了两个关于强模式避免的计数结果.我们也考虑排列的权力都避免了一个模式$τ$。最后，我们研究了对称群的子群，其元素都避免某些模式。这导致了几个新的开放的问题连接的对称群的群结构与模式避免。

英文摘要：

Recently, Bóna and Smith defined strong pattern avoidance, saying that a permutation $π$ strongly avoids a pattern $τ$ if $π$ and $π^2$ both avoid $τ$. They conjectured that for every positive integer $k$, there is a permutation in $S_{k^3}$ that strongly avoids $123\cdots (k+1)$. We use the Robinson--Schensted--Knuth correspondence to settle this conjecture, showing that the number of such permutations is at least $k^{k^3/2+O(k^3/\log k)}$ and at most $k^{2k^3+O(k^3/\log k)}$. We enumerate $231$-avoiding permutations of order $3$, and we give two further enumerative results concerning strong pattern avoidance. We also consider permutations whose powers all avoid a pattern $τ$. Finally, we study subgroups of symmetric groups whose elements all avoid certain patterns. This leads to several new open problems connecting the group structures of symmetric groups with pattern avoidance.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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