Pattern-avoiding shallow permutations

TL;DR

This paper studies shallow permutations, focusing on those avoiding patterns of length 3, and develops structural results to enumerate such permutations, extending understanding of their properties and classifications.

Contribution

It introduces a comprehensive framework for analyzing pattern-avoiding shallow permutations and provides enumeration results for all patterns of length 3.

Findings

Shallow permutations avoiding 321 are characterized as Boolean permutations.

Structural results enable enumeration of all pattern-avoiding shallow permutations.

The paper extends known properties of 321-avoiding shallow permutations to all patterns of length 3.

Abstract

Shallow permutations were defined in 1977 to be those that satisfy the lower bound of the Diaconis-Graham inequality. Recently, there has been renewed interest in these permutations. In particular, Berman and Tenner showed they satisfy certain pattern avoidance conditions in their cycle form and Woo showed they are exactly those whose cycle diagrams are unlinked. Shallow permutations that avoid 321 have appeared in many contexts; they are those permutations for which depth equals the reflection length, they have unimodal cycles, and they have been called Boolean permutations. Motivated by this interest in 321-avoiding shallow permutations, we investigate $\sigma$-avoiding shallow permutations for all $\sigma \in \mathcal{S}_3$. To do this, we develop more general structural results about shallow permutations, and apply them to enumerate shallow permutations avoiding any pattern of length 3.