Rooted forests that avoid sets of permutations

TL;DR

This paper studies classes of rooted forests that avoid specific permutation patterns, providing enumeration results and exploring an analog of Wilf-equivalence for these forests.

Contribution

It introduces a novel permutation-avoidance framework for rooted forests and establishes enumeration and bijection results, including an analog of Wilf-equivalence.

Findings

Enumeration of unimodal forests avoiding certain permutations

Bijections between forests and set partitions with specific properties

Definition and analysis of Wilf-equivalence analog for forests

Abstract

We say that an unordered rooted labeled forest avoids the pattern $\pi\in\mathcal{S}_n$ if the sequence obtained from the labels along the path from the root to any vertex does not contain a subsequence that is in the same relative order as $\pi$. We enumerate several classes of forests that avoid certain sets of permutations, including the set of unimodal forests, via bijections with set partitions with certain properties. We also define and investigate an analog of Wilf-equivalence for forests.