Recently, Bona and Smith defined $\textit{strong pattern avoidance}$, saying that a permutation $\pi$ strongly avoids a pattern $\tau$ if $\pi$ and $\pi^2$ both avoid $\tau$. 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 $2^{k^3+O(k^2\log k)}$. 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 $\textit{all}$ avoid a pattern $\tau$. 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.

中文翻译：

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避免模​​式的排列幂

最近，Bona 和 Smith 定义了 $\textit{strong pattern avoidance}$，说如果 $\pi$ 和 $\pi^2$ 都避免 $\tau$，则置换 $\pi$ 强烈避免模式 $\tau$ . 他们推测，对于每一个正整数 $k$，$S_{k^3}$ 中都有一个置换强烈避开 $123\cdots (k+1)$。我们使用 Robinson--Schensted--Knuth 对应来解决这个猜想，表明这种排列的数量至少为 $2^{k^3+O(k^2\log k)}$。我们枚举了 $231$-避免 $3$ 订单的排列，并且我们给出了两个关于强模式避免的进一步枚举结果。我们还考虑了其​​幂 $\textit{all}$ 避免模式 $\tau$ 的排列。最后，我们研究对称群的子群，它们的元素都避免某些模式。