The Latin Tableau Conjecture

TL;DR

This paper investigates the Latin tableau conjecture, providing computational evidence and partial proofs, advancing understanding of conditions for Latin tableaux existence based on shape and type.

Contribution

It offers computational support for the conjecture and proves its validity for the first four parts of the type vector for any shape.

Findings

Computational evidence supports the conjecture.

The conjecture is proven for the first four parts of the type vector.

Provides insights into conditions for Latin tableau existence.

Abstract

A Latin tableau of shape $\lambda$ and type $\mu$ is a Young diagram of shape $\lambda$ in which each box contains a single positive integer, with no repeated integers in any row or column, and the $i$th most common integer appearing $\mu_i$ times. Over twenty years ago, Chow et al., in their study of a generalization of Rota's basis conjecture that they called the wide partition conjecture, conjectured a necessary and sufficient condition for the existence of a Latin tableau of shape $\lambda$ and type $\mu$. We report some computational evidence for this conjecture, and prove that the conjecture correctly characterizes, for any given $\lambda$, at least the first four parts of $\mu$.