# Pattern avoidance in permutations and their squares

## TL;DR

This paper investigates permutations avoiding a pattern and their squares avoiding the same pattern, providing generating functions, bounds, and open questions for specific patterns.

## Contribution

It introduces new results on pattern avoidance in permutations and their squares, including generating functions and bounds for specific patterns.

## Key findings

- Generated a function for pattern 312 avoidance in permutations and their squares.
- Proved no such permutations exist beyond a certain length for monotone increasing patterns.
- Provided an upper bound for pattern 321 avoidance in permutations and their squares.

## Abstract

We study permutations $p$ such that both $p$ and $p^2$ avoid a given pattern $q$. We obtain a generating function for the case of $q=312$ (equivalently, $q=231$), we prove that if $q$ is monotone increasing, then above a certain length, there are no such permutations, and we prove an upper bound for $q=321$. We also present some intriguing questions in the case of $q=132$.