Outline Rectangles, Allocations, and Latin Young Diagrams
Jack Allsop, Daniel Kotlar, Ian Wanless
TL;DR
The paper connects wide Young diagrams to Latin fillings via allocations and Hilton's outline-rectangle method, proving that a diagram has an allocation if and only if it is Latin. This equivalence reduces Chow et al.'s Wide Partition Conjecture to showing wide diagrams admit allocations, and the authors establish the conjecture for diagrams with three distinct row lengths. It also provides a practical wideness test with quadratic complexity and a constructive proof for the three-length case, while outlining generalization strategies through allocations and a hypergraph framework. Collectively, the work advances the understanding of Latin fillings in the free matroid setting and suggests inductive and hypergraph-based routes toward the general conjecture.
Abstract
A Young diagram is \emph{Latin} if there is an assignment of integers to its cells so that each row $i$ of length $l_i$ is populated by the numbers $1,\ldots,l_i$, and the numbers in each column are distinct. A Young diagram is called \emph{wide} if any subdiagram, formed by a subset of its rows, dominates its conjugate. Chow et al. [Advances in Applied Mathematics, 31, 2003] conjectured that any wide Young diagram is Latin. We introduce a notion of an \emph{allocation} which can be thought of as a coarse attempt at finding a Latin filling for a Young diagram. Using a theorem of Hilton, we prove that a Young diagram has an allocation if and only if it is Latin. This enables us to prove Chow et al.'s conjecture for Young diagrams with three distinct row lengths.