Pattern-Avoiding Permutation Powers

Recently, B'ona and Smith defined 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 $k^{k^3\mathrm{/}2+O(k^3\mathrm{/}\log k)}$ and at most $k^{2k^3+O(k^3\mathrm{/}\log k)}$. 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 $\tau$. 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.