Pattern distributions in Dyck paths with a first return decomposition constrained by height

TL;DR

This paper derives generating functions for pattern distributions in Dyck paths with height constraints, focusing on patterns of length up to three, enhancing understanding of their combinatorial structure.

Contribution

It introduces explicit generating functions for pattern popularity and distribution in height-constrained Dyck paths, a novel combinatorial analysis.

Findings

Explicit generating functions for pattern distributions

Analysis of pattern popularity in constrained Dyck paths

Enhanced understanding of combinatorial structures in Dyck paths

Abstract

We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.