# Decompositions of Partially Ordered Sets

DOI: 10.1023/A:1006420120193

- Cite this article as:
- Billera, L.J. & Hetyei, G. Order (2000) 17: 141. doi:10.1023/A:1006420120193

## Abstract

Generalizing the proof of the theorem describing the closed cone of flag *f*-vectors of arbitrary graded posets, we give a description of the cone of flag *f*-vectors of planar graded posets. The labeling used is a special case of a “chain-edge labeling with the first atom property”, or FA-labeling, which also generalizes the notion of lexicographic shelling, or CL-labeling. The resulting analogy suggests a planar analogue of the flag *h*-vector. For planar Cohen–Macaulay posets the two *h*-vectors turn out to be equal, and the cone of flag *h*-vectors an orthant, whose dimension is a Fibonacci number. The use of FA-labelings also yields a simple enumeration of the facets in the order complex of an arbitrary graded poset such that the intersection of each cell with the previously attached cells is homotopic to a ball or to a sphere.

*f*-vectorlatticeleveled planarlexicographically shellablepartially ordered setplanarshellable