A complete h-vector for convex polytopes

摘要

This note defines a complete h-vector for convex polytopes, which extends thealready known toric (or mpih) h-vector and has many similar properties.Complete means that it encodes the whole of the flag vector. First we define the concept of a generalised h-vector and state someproperties that follow. The toric h-vector is given as an example. We thendefine a complete generalised h-vector, and again state properties. Finally, weshow that this complete h-vector and all with similar properties will sometimeshave negative coefficients. Most of the proofs, and further investigations, will appear elsewhere.