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 $\mathcal{L}(P)$ 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 | ${f}_{S}$ |
---|---|
$\mathrm{\varnothing}$ | 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^{} ${F}_{d-1}$.
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 |
---|---|
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 |