PlanetMath: f-vector

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f-vector

(Definition)

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:

$\displaystyle \sum_{-1\le i\le d}(-1)^i f_i=0.$

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

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
$\emptyset$	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 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 -vector of a d-polytope is one less than the Fibonacci number $F_{d-1}$ .

Bibliography

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.