From Dyck paths to standard Young tableaux

TL;DR

This paper introduces nine bijections connecting Dyck paths, Motzkin paths, and standard Young tableaux, providing new combinatorial descriptions and a unified framework for these structures.

Contribution

It presents novel bijections between Dyck paths, Motzkin paths, and SYT, including a new class of labeled Dyck paths and a dual set partition framework.

Findings

Nine bijections between Dyck paths and SYT classes

Introduction of a new class of labeled Dyck paths

Bijections from Motzkin paths to SYT

Abstract

We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength $n$ that is shown to be in bijection with the set of all SYT with $n$ boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of noncrossing partitions.