Quite recently, we found a new proof of the first non-trivial case of the following result of R. Stanley. We prove this result for the special case where there are exactly three non-zero terms in the sequence. Despite its specialized nature, our proof gives valuable structural information and holds hope of being extended to the general case. It also strengthens a result of Lickorish about non-shellable spheres. The problem of characterizing the f-vectors of all polytopes has proved intractable.
More combinatorial information is encoded in the flag vector, a useful extension of the f-vector.
In this talk I will define flag vectors, the cd-index and Eulerian posets, and will survey results on linear inequalities. The model for a polymatroid is a collection of subspaces, or in the dual space, a subspace arrangement. Many of the results and methods for matroids do not work for polymatroids. In this talk, we will discuss the similarities and differences between the critical problem for matroids and the critical problem for polymatroids. In particular, we will develop a uniform geometric approach for constructing blocks and tangential blocks.