fvector

An fvector, often written f-vector, is a sequence that records the numbers of faces of different dimensions in a polytope (or, more generally, in a simplicial complex). For a d-dimensional polytope P, the f-vector is f(P) = (f0, f1, ..., f(d−1)), where fi denotes the number of i-dimensional faces of P. In this convention, i runs from 0 (vertices) up to d−1 (facets). Some authors include an additional entry f_d = 1 to count the polytope itself, but the standard modern convention for polytopes uses up to f(d−1).

Examples help illustrate the concept. A tetrahedron has 4 vertices, 6 edges, and 4 triangular faces, so its f-vector is (4, 6, 4). A cube has (8, 12, 6). A 4-simplex has (5, 10, 10, 5). These vectors are invariant under combinatorial equivalence, meaning isomorphic face-incidence structures yield the same f-vector.

Duality and relations. The f-vector of the dual polytope P* is obtained by reversing indices: f_i(P*) = f_{d−1−i}(P) for i = 0, ..., d−1. In three dimensions, convex polyhedra satisfy Euler’s formula f0 − f1 + f2 = 2. Beyond specific cases, the f-vector interacts with other encodings of combinatorial structure, such as the h-vector, which is obtained from the f-vector by a linear transformation that often simplifies combinatorial identities.

Extensions. The concept extends to simplicial complexes and triangulations, where fi counts the number of i-dimensional faces in the complex. The f-vector provides a compact, essential snapshot of a shape’s combinatorial skeleton and is central to enumerative combinatorics and polytope theory.