Pattern avoidance and the fundamental bijection

TL;DR

This paper studies permutations avoiding patterns in S_3 that remain avoiding under the fundamental bijection and its iterations, providing enumeration results and exploring fixed points.

Contribution

It introduces new enumeration results for pattern-avoiding permutations under the fundamental bijection and its iterations, extending understanding of permutation pattern avoidance.

Findings

Enumerates permutations avoiding a pattern and its image under the bijection

Analyzes permutations fixed under the second iteration of the bijection

Provides directions for future research in permutation pattern avoidance

Abstract

The fundamental bijection is a bijection $\theta:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations $\pi \in \mathcal{S}_n$ that avoids a pattern $\sigma \in \mathcal{S}_3$, whose image $\theta(\pi)$ also avoids $\sigma$. We additionally consider what happens under repeated iterations of $\theta$; in particular, we enumerate permutations $\pi \in \mathcal{S}_n$ that have the property that $\pi$ and its first $k$ iterations under $\theta$ all avoid a pattern $\sigma$. Finally, we consider permutations with the property that $\pi=\theta^2(\pi)$ that avoid a given pattern $\sigma$, and end the paper with some directions for future study.