Pattern avoidance in forests of binary shrubs

TL;DR

This paper studies pattern avoidance in permutations constrained by forest-like structures called binary shrub forests, providing explicit enumerations and growth rates for various pattern-avoidance cases.

Contribution

It introduces a novel framework linking pattern avoidance in permutations to binary shrub forests and offers explicit enumeration results for multiple cases.

Findings

Explicit enumeration for four pattern-avoidance cases via lattice path bijections.

Determination of growth rate for the remaining case using analytic combinatorics.

Identification of connections to well-known lattice paths like Duchon's club paths.

Abstract

We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.