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Partitons of vertices and facets in trees and stacked simplicial complexes

Gunnar Fløystad

Abstract

For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of vertices into independent sets. More generally we give bijections between facet partitions whose parts have minimal distance $\geq s$ and vertex partitions whose parts have minimal distance $\geq s+1$. A consequence is results on partitions of natural numbers, where the parts have minimal bounds on spacing.

Partitons of vertices and facets in trees and stacked simplicial complexes

Abstract

For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of vertices into independent sets. More generally we give bijections between facet partitions whose parts have minimal distance and vertex partitions whose parts have minimal distance . A consequence is results on partitions of natural numbers, where the parts have minimal bounds on spacing.
Paper Structure (12 sections, 15 theorems, 59 equations, 2 figures)

This paper contains 12 sections, 15 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stacked simplicial complexes
  3. Paths
  4. Existence and uniqueness of paths
  5. Distance neighborhoods
  6. Bijections between partitions of facets and of vertices
  7. Bijections between partitions
  8. From vertex partitions to facet partitions.
  9. From facet partitions to vertex partitions
  10. Induction arguments on $X_m$
  11. The main theorem
  12. Partitions of natural numbers

Key Result

Lemma 2.5

Let $X$ be a stacked simplicial complex and $f_1,f_2, \ldots, f_p$ a path in $X$.

Figures (2)

  • Figure 1: Partitions of edges into two parts and corresponding partitions of vertices into three parts
  • Figure 2: Partition of facets into two parts and corresponding partition of vertices into four parts

Theorems & Definitions (39)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 29 more