Let P be a polytope of dimension d. The f-vector of P is the finite integer sequence (f0,,fd-i), where the componentMathworldPlanetmath in position i is the number of i-dimensional faces of P. For some purposes it is convenient to view the empty face and the polytope itself as improper faces, so f-1=fd=1.

For example, a cube has 8 vertices, 12 edges, and 6 faces, so its f-vector is (8, 12, 6).

The entries in the f-vector of a convex polytope satisfy the Euler–Poincaré–Schläfli formula:


Consequently, the face latticePlanetmathPlanetmath of a polytope is Eulerian. For any graded poset with maximum and minimum elements there is an extension of the f-vector called the flag f-vector. For any subset S of {0,1,,d-1}, the fS entry of the flag f-vector of P is the number of chains of faces in (P) with dimensions coming only from S.

The flag f-vector of a three-dimensional cube is given in the following table. For simplicity we drop braces and commas.

S fS
0 8
1 12
2 6
01 83=24
02 83=24
12 122=24
012 832=48

For example, f{1,2}=24 because each of the 12 edges meets exactly two faces.

Although the flag f-vector of a d-polytope has 2d entries, most of them are redundant, as they satisfy a collection of identitiesPlanetmathPlanetmathPlanetmath generalizing the Euler–Poincaré–Schläfli formula and called the generalized Dehn-Sommerville relations. Interestingly, the number of nonredundant entries in the flag f-vector of a d-polytope is one less than the Fibonacci numberMathworldPlanetmath Fd-1.


  • 1 Bayer, M. and L. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered setsMathworldPlanetmath, Invent. Math. 79 (1985), no. 1, 143–157.
  • 2 Bayer, M. and A. Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47.
  • 3 Ziegler, G., Lectures on polytopes, Springer-Verlag, 1997.
Title f-vector
Canonical name Fvector
Date of creation 2013-03-22 16:59:10
Last modified on 2013-03-22 16:59:10
Owner mps (409)
Last modified by mps (409)
Numerical id 5
Author mps (409)
Entry type Definition
Classification msc 52B40
Synonym f-vector
Defines flag f-vector
Defines flag f-vector