---
tag: DCK64GFPD7
kind: concept
name: f-vector and h-vector
msc: ["05E45", 52B05]
sources:
  - url: https://en.wikipedia.org/wiki/H-vector
---

## Summary

The f-vector of a [[simplicial complex|V8GZ0W1ZZ5]] or of a polytope lists the number of its faces of each dimension.
The h-vector is an invertible linear transform of the f-vector that often has simpler properties.
It is symmetric for simplicial spheres and nonnegative for many important complexes.
The [[Dehn–Sommerville equations|4WKHF5BCTA]] and the [[g-theorem|4CZMWQVC30]] are stated in terms of the h-vector.

## Explanations

### L1 human

Count the vertices, edges, triangles, and so on of a shape built from simplices, and you get its f-vector.
For the surface of an octahedron the counts are $6$ vertices, $12$ edges and $8$ triangles.
Rewriting these numbers as an h-vector gives $(1, 3, 3, 1)$, which reads the same forwards and backwards.
This hidden symmetry is one reason to use the h-vector.