f-vector

Let P be a polytope of dimension d. The f-vector of P is the finite integer sequence $(f_0,\mathrm{\dots },f_{d-i})$, 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:

$$\sum _{-1\le i\le d}{(-1)}^i{f}_i=0.$$

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 $\{0,1,\mathrm{\dots },d-1\}$, the $f_S$ entry of the flag f-vector of P is the number of chains of faces in $ℒ(P)$ 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	$f_S$
$\mathrm{\varnothing }$	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 $F_{d-1}$.

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