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.
chain Cohen–Macaulay EL-labeling flag flag f-vector lattice leveled planar lexicographically shellable partially ordered set planar shellable