Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions

TL;DR

This paper establishes a combinatorial correspondence between pairs of noncrossing free Dyck paths and noncrossing partitions, providing new insights into their structural relationships using bijections and algorithms.

Contribution

It introduces a novel bijection linking pairs of noncrossing free Dyck paths to noncrossing partitions, and characterizes Dyck paths via the Labelle merging algorithm.

Findings

Established a bijection between path pairs and partitions

Characterized Dyck paths using the Labelle merging algorithm

Connected path structures with partition properties

Abstract

Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.