Noncrossing Linked Partitions and Large (3,2)-Motzkin Paths

TL;DR

This paper establishes a bijection between noncrossing linked partitions and large (3,2)-Motzkin paths, providing combinatorial insights into free probability and Schroder numbers.

Contribution

It introduces a novel correspondence linking noncrossing linked partitions with large (3,2)-Motzkin paths, elucidating their combinatorial structure.

Findings

Established a one-to-one correspondence between noncrossing linked partitions and large (3,2)-Motzkin paths.

Provided a combinatorial explanation for the relation between large and little Schroder numbers.

Connected free probability concepts with combinatorial path models.

Abstract

Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and colored Motzkin paths. A (3,2)-Motzkin path can be viewed as a colored Motzkin path in the sense that there are three types of level steps and two types of down steps. A large (3,2)-Motzkin path is defined to be a (3,2)-Motzkin path for which there are only two types of level steps on the x-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of [n+1] and the set of large (3,2)-Motzkin paths of length n. In this setting, we get a simple explanation of the well-known relation between the large and the little Schroder numbers.