Words, Dyck paths, Trees, and Bijections

TL;DR

This paper explores the combinatorial structures of nondecreasing Dyck paths by establishing bijections with planted plane trees and analyzing related statistics, extending previous work with new connections to words, rational languages, and continued fractions.

Contribution

It introduces a new bijection between nondecreasing Dyck paths and planted plane trees of height ≤ 4, enriching the combinatorial understanding of these objects.

Findings

Established a bijection with planted plane trees of height ≤ 4

Computed new statistics on trees corresponding to Dyck paths

Extended the framework to words, rational languages, and continued fractions

Abstract

In \cite{BaDeFePi96} the concept of nondecreasing Dyck paths was introduced. We continue this research by looking at it from the point of view of words, rational languages, planted plane trees, and continued fractions. We construct a bijection with planted plane trees of height $\le 4$ and compute various statistics on trees that are the equivalents of nondecreasing Dyck paths.