Standard Young Tableaux and Colored Motzkin Paths

TL;DR

This paper introduces colored Motzkin paths and establishes a bijection with standard Young tableaux of bounded height, providing new combinatorial interpretations and revealing intrinsic relations between tableaux with different row bounds.

Contribution

It presents a novel bijection between standard Young tableaux and colored Motzkin paths, linking two combinatorial objects and uncovering new structural insights.

Findings

Bijection between SYT and colored Motzkin paths

Lattice path interpretation of SYT

Intrinsic relation between SYT with different row bounds

Abstract

In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the $n$-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length $n$. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most $2d+1$ rows and the set of SYTs with at most 2d rows.