Classical and consecutive pattern avoidance in rooted forests

TL;DR

This paper explores pattern avoidance in rooted forests, establishing bijections, defining forest-Young diagrams, and extending the cluster method to analyze and classify consecutive pattern avoidance, revealing new equivalences.

Contribution

It introduces forest-Young diagrams, generalizes pattern avoidance results, and extends the cluster method to rooted forests, providing new insights into pattern equivalences.

Findings

Bijection between forests avoiding specific patterns

Enumeration recurrences for certain pattern sets

Extension of the cluster method to forests

Abstract

Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern $\sigma\in\mathcal{S}_k$ if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as $\sigma$. For each permutation $\sigma\in\mathcal{S}_{k-2}$, we construct a bijection between $n$-vertex forests avoiding $(\sigma)(k-1)k:=\sigma(1)\cdots\sigma(k-2)(k-1)k$ and $n$-vertex forests avoiding $(\sigma)k(k-1):=\sigma(1)\cdots\sigma(k-2)k(k-1)$, giving a common generalization of results of West on permutations and Anders--Archer on forests. We further define a new object, the forest-Young diagram, which we use to extend the notion of shape-Wilf equivalence to forests. In particular, this allows us to generalize the above result to a bijection between forests avoiding $\{(\sigma_1)k(k-1), (\sigma_2)k(k-1), \dots, (\sigma_\ell)k(k-1)\}$ and forests avoiding $\{(\sigma_1)(k-1)k, (\sigma_2)(k-1)k, \dots, (\sigma_\ell)(k-1)k\}$ for $\sigma_1, \dots, \sigma_\ell \in \mathcal{S}_{k-2}$. Furthermore, we give recurrences enumerating the forests avoiding $\{123\cdots k\}$, $\{213\}$, and other sets of patterns. Finally, we extend the Goulden--Jackson cluster method to study consecutive pattern avoidance in rooted trees as defined by Anders and Archer. Using the generalized cluster method, we prove that if two length-$k$ patterns are strong-c-forest-Wilf equivalent, then up to complementation, the two patterns must start with the same number. We also prove the surprising result that the patterns $1324$ and $1423$ are strong-c-forest-Wilf equivalent, even though they are not c-Wilf equivalent with respect to permutations.