# Pattern-Avoiding Permutation Powers

## TL;DR

This paper investigates strong pattern avoidance in permutations, proving conjectures, providing enumeration results, and exploring algebraic structures related to pattern-avoiding permutations.

## Contribution

It settles a conjecture on strong pattern avoidance using the RSK correspondence and offers new enumeration and structural results.

## Key findings

- Number of strongly avoiding permutations is at least k^{k^3/2+O(k^3/ ext{log} k)}
- Number of such permutations is at most k^{2k^3+O(k^3/ ext{log} k)}
- Enumerates 231-avoiding permutations of order 3 and explores related pattern avoidance properties.

## Abstract

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 $k^{k^3/2+O(k^3/\log k)}$ 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 $\tau$. 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.