Outline Rectangles, Allocations, and Latin Young Diagrams
Jack Allsop, Daniel Kotlar, Ian Wanless

TL;DR
This paper proves a conjecture that wide Young diagrams are Latin by introducing allocations and using Hilton's theorem, confirming the conjecture for diagrams with three distinct row lengths.
Contribution
It establishes a new equivalence between allocations and Latin fillings in Young diagrams and proves the conjecture for diagrams with three distinct row lengths.
Findings
A Young diagram has an allocation if and only if it is Latin.
The conjecture holds for Young diagrams with three distinct row lengths.
Introduces the concept of allocations as a tool for studying Latin diagrams.
Abstract
A Young diagram is \emph{Latin} if there is an assignment of integers to its cells so that each row of length is populated by the numbers , 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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Random Matrices and Applications · Markov Chains and Monte Carlo Methods
