Partitons of vertices and facets in trees and stacked simplicial complexes

TL;DR

This paper establishes explicit bijections between facet partitions and vertex partitions with minimal distance constraints in stacked simplicial complexes, including trees and polytopes, leading to new results on number partitions with spacing conditions.

Contribution

It introduces a novel combinatorial bijection framework connecting facet and vertex partitions with minimal distance constraints in stacked simplicial complexes.

Findings

Bijections between facet and vertex partitions with minimal distance constraints

Results on partitions of natural numbers with spacing bounds

Applications to trees, polygons, and polytopes

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.