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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2603.21221 |
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| _version_ | 1866908930030960640 |
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| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | For each positive integer $n$, let $G_n$ be the graph of integer partitions of $n$, where two partitions are adjacent if one is obtained from the other by an elementary transfer of a cell in the Ferrers diagram, followed by reordering. Previous work has studied the global homotopy type of the clique complex $Cl(G_n)$ and the local combinatorics of $G_n$ at a fixed vertex. This paper initiates the study of $G_n$ itself as a growing discrete geometric object. It introduces a structural language for the large-scale morphology of partition graphs, centered on the antenna vertices, main chain, boundary framework, self-conjugate axis, simplex layers, degree landscape, central region, and spine. Using local invariants from the companion local theory, it also defines canonical vertex layerings of $G_n$. A small computational atlas for $1 \le n \le 12$ is included to illustrate how these structures emerge and interact. The paper is intended as a foundational and exploratory contribution, providing a vocabulary, a first structural picture, and a set of open directions for future quantitative and asymptotic work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Partition Graph as a Growing Discrete Geometric Object Lyudogovskiy, Fedor B. General Mathematics 05A17, 05C75, 05E10 For each positive integer $n$, let $G_n$ be the graph of integer partitions of $n$, where two partitions are adjacent if one is obtained from the other by an elementary transfer of a cell in the Ferrers diagram, followed by reordering. Previous work has studied the global homotopy type of the clique complex $Cl(G_n)$ and the local combinatorics of $G_n$ at a fixed vertex. This paper initiates the study of $G_n$ itself as a growing discrete geometric object. It introduces a structural language for the large-scale morphology of partition graphs, centered on the antenna vertices, main chain, boundary framework, self-conjugate axis, simplex layers, degree landscape, central region, and spine. Using local invariants from the companion local theory, it also defines canonical vertex layerings of $G_n$. A small computational atlas for $1 \le n \le 12$ is included to illustrate how these structures emerge and interact. The paper is intended as a foundational and exploratory contribution, providing a vocabulary, a first structural picture, and a set of open directions for future quantitative and asymptotic work. |
| title | The Partition Graph as a Growing Discrete Geometric Object |
| topic | General Mathematics 05A17, 05C75, 05E10 |
| url | https://arxiv.org/abs/2603.21221 |