On the three-rowed skew standard Young tableaux

TL;DR

This paper introduces a bijection between Motzkin paths and three-rowed standard Young tableaux, providing a uniform combinatorial proof for identities involving skew shapes and Motzkin numbers.

Contribution

It establishes a simple bijection that offers a uniform combinatorial proof for identities relating skew Young tableaux and Motzkin numbers.

Findings

Confirmed Zeilberger's conjecture with a bijective proof.

Unified approach to identities involving Motzkin numbers and Young tableaux.

Provided a combinatorial interpretation for a class of Motzkin-related identities.

Abstract

Let $\mathcal{T}_3$ be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with $n-3$ entries in the "skew three-rowed strip" $\mathcal{T}_3 / (2,1,0)$ is $m_{n-1}-m_{n-3}$, a difference of two Motzkin numbers. This conjecture, together with hundreds of similar identities, were derived automatically and proved rigorously by Zeilberger via his powerful program and WZ method. It appears that each one is a linear combination of Motzkin numbers with constant coefficients. In this paper we will introduce a simple bijection between Motzkin paths and standard Young tableaux with at most three rows. With this bijection we answer Zeilberger's question affirmatively that there is a uniform way to construct bijective proofs for all of those identites.