Partitions 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 with their triangulations), 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 \(\ge s\) and vertex partitions whose parts have minimal distance \(\ge s+1\).