A brief introduction to the h-vector and its specializations:

The f-vector of a simplicial polytope (or complex) is a vector enumerating its number of faces of each dimensions. We may linearly transform this to the h-vector, which maintains the original combinatorial data of the f-vector while making explicit the underlying linear relations between face numbers. It has a rich mathematical structure that has inspired research for the last fifty years. Though less intuitive than the unaltered face numbers, it often lets us conjecture about face numbers of polytopes in more elegant ways than the f-vector does. In particular, the celebrated Lower Bound Theorem that characterizes the f-vectors of all simplicial polytopes is typically written using the h-vector and its variants.

In order to extend the h-vector beyond simplicial geometry and to all Eulerian posets, Stanley developed the toric h-vector. The toric h-vector makes explicit the relationship between polytope theory and algebraic geometry, as well as lets us use h-vector techniques on a significantly larger class of objects than simplicial polytopes. As the study of geometric and poset subdivisions became more relevant, the local h-vector was developed to express decompositions of a subdivided complex relative to its original structure. The theory developed while studying local h-vectors has also been significant in the development of Kahzdan-Lusztig polynomials of Coxeter groups. Recently, Katz and Stapledon demonstrated that the local h-vector can be specialized even further into a mixed h-polynomial in two variables.

We will examine these vectors and try to explain how they work and why they're useful to combinatorialists. Time permitting, we may even discuss algebra morphisms between enumerating flags and enumerating face vectors.