f-vector

Let P be a polytope of dimension d. The f-vector ofP is the finite integer sequence $(f_0,\mathrm{\dots },f_{d-i})$, wherethe component in position i is the number of i-dimensionalfaces of P. For some purposes it is convenient to view the emptyface 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 itsf-vector is (8, 12, 6).

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

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

Consequently, the face lattice of a polytope is Eulerian. For anygraded poset with maximum and minimum elements there is an extensionof the f-vector called the flag f-vector. For any subsetS of $\{0,1,\mathrm{\dots },d-1\}$, the $f_S$ entry of the flagf-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 thefollowing 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 edgesmeets 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 identitiesgeneralizing the Euler–Poincaré–Schläfli formula and called thegeneralized Dehn-Sommerville relations. Interestingly, the number ofnonredundant entries in the flag $f$-vector of a d-polytope isone less than the Fibonacci number $F_{d-1}$.