Partitions of Vertices and Facets in Trees and Stacked Simplicial Complexes

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 \(\ge s\) $\ge s$ and vertex partitions whose parts have minimal distance \(\ge s+1\) $\ge s+1$ .