A complete h-vector for convex polytopes

TL;DR

This paper introduces a complete h-vector for convex polytopes that encodes the entire flag vector, extending existing concepts and revealing new properties including potential negative coefficients.

Contribution

It defines a new complete h-vector for convex polytopes that generalizes the toric h-vector and encodes the full flag vector, with initial properties and implications.

Findings

Complete h-vector encodes the entire flag vector.

The complete h-vector can have negative coefficients.

Properties similar to the toric h-vector are established.

Abstract

This note defines a complete h-vector for convex polytopes, which extends the already 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 some properties that follow. The toric h-vector is given as an example. We then define a complete generalised h-vector, and again state properties. Finally, we show that this complete h-vector and all with similar properties will sometimes have negative coefficients. Most of the proofs, and further investigations, will appear elsewhere.