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 $\geq s$ 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.