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The Partition Graph as a Growing Discrete Geometric Object

Fedor B. Lyudogovskiy

Abstract

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.

The Partition Graph as a Growing Discrete Geometric Object

Abstract

For each positive integer , let be the graph of integer partitions of , 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 and the local combinatorics of at a fixed vertex. This paper initiates the study of 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 . A small computational atlas for 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.
Paper Structure (54 sections, 6 theorems, 60 equations, 13 figures, 3 tables)

This paper contains 54 sections, 6 theorems, 60 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The partition graph as a geometric object
  3. The graph G_n
  4. Ferrers-diagram viewpoint
  5. The antenna vertices
  6. The main chain
  7. The left and right edges
  8. Boundary framework
  9. Conjugation symmetry and the self-conjugate axis
  10. Conjugation as an automorphism
  11. The self-conjugate axis
  12. Axial distance
  13. Local simplex layers and morphological organization
  14. Local input from the companion paper
  15. Simplex layers
  16. ...and 39 more sections

Key Result

Lemma 2.1

For every $n\ge 1$, the graph $G_n$ is connected.

Figures (13)

  • Figure 1: Structural atlas, Part I: $G_1,\dots,G_4$. Boundary-framework vertices are lightly tinted, interior vertices are white, self-conjugate vertices are dark blue, and the dashed line marks the self-conjugate axis.
  • Figure 2: Structural atlas, Part II: $G_5,\dots,G_8$.
  • Figure 3: Structural atlas, Part III: $G_9,\dots,G_{12}$.
  • Figure 4: Degree atlas, Part I: $G_1,\dots,G_4$. Node color indicates degree, with a common scale shared across all degree-atlas pages.
  • Figure 5: Degree atlas, Part II: $G_5,\dots,G_8$.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • ...and 16 more