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Axial Morphology of the Partition Graph: Self-Conjugate Axis, Spine, and Concentration

Fedor B. Lyudogovskiy

Abstract

We study the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts. We define the self-conjugate axis, its distance neighborhoods, and the thin spine, a first off-axis layer built from common neighbors of distinct axial vertices. We prove that distinct self-conjugate vertices are never adjacent, that the thin spine is a conjugation-invariant induced subgraph, and that axial and spinal concentration radii differ by at most one. Computations for $1 \le n \le 30$ show that the main local invariants are maximized near the axis and the spine.

Axial Morphology of the Partition Graph: Self-Conjugate Axis, Spine, and Concentration

Abstract

We study the partition graph , whose vertices are the partitions of and whose edges correspond to elementary unit transfers between parts. We define the self-conjugate axis, its distance neighborhoods, and the thin spine, a first off-axis layer built from common neighbors of distinct axial vertices. We prove that distinct self-conjugate vertices are never adjacent, that the thin spine is a conjugation-invariant induced subgraph, and that axial and spinal concentration radii differ by at most one. Computations for show that the main local invariants are maximized near the axis and the spine.
Paper Structure (53 sections, 18 theorems, 95 equations, 2 figures)

This paper contains 53 sections, 18 theorems, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Axial Definitions
  3. The partition graph and conjugation
  4. The self-conjugate axis
  5. Axial distance and central regions
  6. Axial interaction graph and mediators
  7. Thin and thick spine
  8. Extremal local complexity and concentration radii
  9. The Self-Conjugate Axis and the Thin Spine
  10. The axis as a fixed-point set
  11. First off-axis bridges between axial vertices
  12. The thin spine
  13. Central Regions, Thick Spines, and Concentration Radii
  14. Axial and spinal filtrations
  15. First comparison between the axis and the spine
  16. ...and 38 more sections

Key Result

Proposition 2.1

The map is an involutive graph automorphism of $G_n$.

Figures (2)

  • Figure 1: Smaller representative examples of axial morphology for $n=5,6,8,9,12,13$.
  • Figure 2: Larger representative examples of axial morphology for $n=19,24,29,30$.

Theorems & Definitions (60)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 50 more