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Simplicial shells and thickness in the partition graph

Fedor B. Lyudogovskiy

Abstract

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.

Simplicial shells and thickness in the partition graph

Abstract

For each positive integer , let be the graph whose vertices are the partitions of , with edges given by elementary transfers of one unit between parts, followed by reordering. We study the local simplex dimension in the clique complex as a geometric thickness invariant of . For a partition , let be its simplicial thickness. This gives threshold thick zones and, relative to the boundary framework of , a shell/core decomposition into outer shells and inner cores . 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 , and the corresponding shell 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 , giving first-occurrence tables for the regimes and supporting a finite-range rear-central thickening pattern.

Paper Structure

This paper contains 48 sections, 20 theorems, 84 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and recalled notation
  3. The partition graph
  4. Local simplex dimension
  5. Conjugation symmetry
  6. Boundary framework
  7. Axis, spine, and central language
  8. Scope of recalled material
  9. Simplicial thickness, threshold zones, shells, and cores
  10. Local simplicial thickness
  11. Threshold thick zones and exact regimes
  12. Outer shells and inner cores
  13. Low-dimensional special cases
  14. Why this choice of language
  15. The exact one-dimensional regime and the boundary framework
  16. ...and 33 more sections

Key Result

Lemma 2.1

For every $n\ge 2$, the partition graph $G_n$ is connected.

Figures (6)

  • Figure 1: Thickness map of $G_4$ at the first triangular threshold $n_2=4$. Vertices are colored by simplicial thickness $\tau_4(\lambda)$, and the maximal-thickness locus $M_4$ is indicated by black outlines. At this stage the only nontrivial thickness value is $\tau=2$, realized by the first triangular regime.
  • Figure 2: Threshold-zone view of $G_7$ at the first tetrahedral threshold $n_3=7$. Gray vertices form the exact one-dimensional regime $T_{=1}(7)$, blue vertices belong to the triangular skin $Sh_2(7)$, and red vertices form the inner tetrahedral core $Core_3(7)$. The maximal-thickness locus $M_7$ is indicated by black outlines; its cardinality is recorded in Table \ref{['tab:max-thickness-loci']}.
  • Figure 3: Thickness map of $G_{11}$ at the first threshold $n_4=11$ for simplicial thickness $4$. The order-$4$ regime is outlined in black and is embedded inside the broader triangular and tetrahedral layers. Its cardinality is recorded in Table \ref{['tab:max-thickness-loci']}.
  • Figure 4: Thickness map of $G_{16}$ at the first threshold $n_5=16$ for simplicial thickness $5$. The maximal-thickness locus is outlined in black and remains localized well inside the triangular skin and away from the front extremes of the framework. Its cardinality is recorded in Table \ref{['tab:max-thickness-loci']}.
  • Figure 5: Thickness map of $G_{22}$ at the first threshold $n_6=22$ for simplicial thickness $6$. The maximal-thickness locus $M_{22}$ is outlined in black. Its cardinality is recorded in Table \ref{['tab:max-thickness-loci']}. Its position is consistent with the same rear-central bias observed at earlier transition levels.
  • ...and 1 more figures

Theorems & Definitions (67)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Definition 2.3
  • Remark 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 57 more