Outline rectangles, allocations, and Latin Young diagrams

A Young diagram is Latin if there is an assignment of integers to its cells so that each row i of length lᵢ is populated by the numbers 1,…,lᵢ, and the numbers in each column are distinct. A Young diagram is called 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 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.