f-vector and h-vector

Summary

The f-vector of a simplicial complex or of a polytope lists the number of its faces of each dimension. The h-vector is an invertible linear transform of the f-vector that often has simpler properties. It is symmetric for simplicial spheres and nonnegative for many important complexes. The Dehn–Sommerville equations and the g-theorem are stated in terms of the h-vector.