f-vector

Summary

The f-vector of a convex polytope lists how many faces it has in each dimension. For a $d$-polytope it is $(f_0, f_1, \dots, f_{d-1})$, where $f_i$ is the number of $i$-dimensional faces. The same notion applies to simplicial complexes, and it is the starting point for the h-vector; see f-vector and h-vector. Describing which vectors are f-vectors is a main problem of polytope combinatorics.