Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.28171 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911559355203584 |
|---|---|
| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges given by elementary transfers of one unit between parts, followed by reordering. We study the local simplex dimension in the clique complex $K_n=\Cl(G_n)$ as a geometric thickness invariant of $G_n$. For a partition $λ\vdash n$, let $τ_n(λ):=\dim_{\mathrm{loc}}(λ)$ be its simplicial thickness. This gives threshold thick zones $T_{\ge r}(n)=\{λ: τ_n(λ)\ge r\}$ and, relative to the boundary framework of $G_n$, a shell/core decomposition into outer shells $Sh_r(n)$ and inner cores $Core_r(n)$. Using local-morphology results established earlier in the series, we work with simplicial thickness as a local invariant. We prove that it is preserved by conjugation, that the induced thick zones, shells, and cores are conjugation-invariant, and that the antennas remain strictly one-dimensional in the simplicial sense and are excluded from all nontrivial thick zones. The first shell order at which a nontrivial shell can occur is therefore $2$, and the corresponding shell $Sh_2(n)$ is the triangular skin, while higher simplicial regimes form nested higher-order shells inside the triangular regime. We also develop a complete finite computational atlas for $1\le n\le 30$, giving first-occurrence tables for the regimes $T_{\ge r}(n)$ and supporting a finite-range rear-central thickening pattern. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28171 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simplicial shells and thickness in the partition graph Lyudogovskiy, Fedor B. General Mathematics 05A17, 05C25, 05C38, 52B05 For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges given by elementary transfers of one unit between parts, followed by reordering. We study the local simplex dimension in the clique complex $K_n=\Cl(G_n)$ as a geometric thickness invariant of $G_n$. For a partition $λ\vdash n$, let $τ_n(λ):=\dim_{\mathrm{loc}}(λ)$ be its simplicial thickness. This gives threshold thick zones $T_{\ge r}(n)=\{λ: τ_n(λ)\ge r\}$ and, relative to the boundary framework of $G_n$, a shell/core decomposition into outer shells $Sh_r(n)$ and inner cores $Core_r(n)$. Using local-morphology results established earlier in the series, we work with simplicial thickness as a local invariant. We prove that it is preserved by conjugation, that the induced thick zones, shells, and cores are conjugation-invariant, and that the antennas remain strictly one-dimensional in the simplicial sense and are excluded from all nontrivial thick zones. The first shell order at which a nontrivial shell can occur is therefore $2$, and the corresponding shell $Sh_2(n)$ is the triangular skin, while higher simplicial regimes form nested higher-order shells inside the triangular regime. We also develop a complete finite computational atlas for $1\le n\le 30$, giving first-occurrence tables for the regimes $T_{\ge r}(n)$ and supporting a finite-range rear-central thickening pattern. |
| title | Simplicial shells and thickness in the partition graph |
| topic | General Mathematics 05A17, 05C25, 05C38, 52B05 |
| url | https://arxiv.org/abs/2603.28171 |