The Latin Tableau Conjecture

Abstract

A Latin tableau of shape $λ$ and type $μ$ is a Young diagram of shape $λ$ 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 $μ_i$ 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 $λ$ and type $μ$. We report some computational evidence for this conjecture, and prove that the conjecture correctly characterizes, for any given $λ$, at least the first four parts of $μ$.