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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.23228 |
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| _version_ | 1866918422810460160 |
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| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | We study the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. We introduce the simplex stratification of $G_n$: for each vertex $λ$, let $\dim_{\mathrm{loc}}(λ)$ denote the largest dimension of a simplex of the clique complex $K_n = \mathrm{Cl}(G_n)$ containing $λ$. This defines a decomposition of $V(G_n)$ into layers $L_r(n)=\{λ\in V(G_n): \dim_{\mathrm{loc}}(λ)=r\}$. We formalize the graph-theoretic interfaces between consecutive layers, called phase boundaries, and study the associated interface graphs and boundary thresholds. Using the previously established star/top description of cliques through a fixed vertex, we show that $\dim_{\mathrm{loc}}(λ)$ is determined exactly by the maximal star and top capacities through $λ$. This yields explicit local criteria for membership in higher simplex layers and reformulates their first appearance in terms of local star/top capacity thresholds. We also present an exhaustive computational study for $n\le 30$, including exact-layer thresholds, boundary thresholds, selected layer profiles, and the behaviour of the boundary framework. The computations suggest a rigid threshold pattern related to staircase partitions and their one-cell extensions, while the corresponding global statements are left as conjectures and open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23228 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simplex Stratification and Phase Boundaries in the Partition Graph Lyudogovskiy, Fedor B. General Mathematics 05A17, 05C69, 05C75, 05E45 We study the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. We introduce the simplex stratification of $G_n$: for each vertex $λ$, let $\dim_{\mathrm{loc}}(λ)$ denote the largest dimension of a simplex of the clique complex $K_n = \mathrm{Cl}(G_n)$ containing $λ$. This defines a decomposition of $V(G_n)$ into layers $L_r(n)=\{λ\in V(G_n): \dim_{\mathrm{loc}}(λ)=r\}$. We formalize the graph-theoretic interfaces between consecutive layers, called phase boundaries, and study the associated interface graphs and boundary thresholds. Using the previously established star/top description of cliques through a fixed vertex, we show that $\dim_{\mathrm{loc}}(λ)$ is determined exactly by the maximal star and top capacities through $λ$. This yields explicit local criteria for membership in higher simplex layers and reformulates their first appearance in terms of local star/top capacity thresholds. We also present an exhaustive computational study for $n\le 30$, including exact-layer thresholds, boundary thresholds, selected layer profiles, and the behaviour of the boundary framework. The computations suggest a rigid threshold pattern related to staircase partitions and their one-cell extensions, while the corresponding global statements are left as conjectures and open problems. |
| title | Simplex Stratification and Phase Boundaries in the Partition Graph |
| topic | General Mathematics 05A17, 05C69, 05C75, 05E45 |
| url | https://arxiv.org/abs/2603.23228 |