# f-vector

Let P be a polytope of dimension d. The f-vector of P is the finite integer sequence $(f_{0},\dots,f_{d-i})$, where the component 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 $f_{-1}=f_{d}=1$.

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:

 $\sum_{-1\leq i\leq d}(-1)^{i}f_{i}=0.$

Consequently, the face lattice 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 $\{0,1,\dots,d-1\}$, the $f_{S}$ entry of the flag f-vector of P is the number of chains of faces in $\mathcal{L}(P)$ 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 $f_{S}$
$\emptyset$ 1
0 8
1 12
2 6
01 $8\cdot 3=24$
02 $8\cdot 3=24$
12 $12\cdot 2=24$
012 $8\cdot 3\cdot 2=48$

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 $2^{d}$ entries, most of them are redundant, as they satisfy a collection of identities 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 number $F_{d-1}$.

## References

• 1 Bayer, M. and L. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, 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 Fvector 2013-03-22 16:59:10 2013-03-22 16:59:10 mps (409) mps (409) 5 mps (409) Definition msc 52B40 $f$-vector flag f-vector flag $f$-vector