Outline Rectangles, Allocations, and Latin Young Diagrams
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909900725026816 |
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| author | Allsop, Jack Kotlar, Daniel Wanless, Ian |
| author_facet | Allsop, Jack Kotlar, Daniel Wanless, Ian |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10548 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Outline Rectangles, Allocations, and Latin Young Diagrams Allsop, Jack Kotlar, Daniel Wanless, Ian Combinatorics 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. |
| title | Outline Rectangles, Allocations, and Latin Young Diagrams |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.10548 |