f-vector and h-vector

Remarks

Comparing coefficients gives $h_k = \sum_{i=0}^{k} (-1)^{k-i} \binom{d-i}{k-i} f_{i-1}$. In particular $h_0 = 1$, $h_1 = f_0 - d$, and $h_0 + h_1 + \dots + h_d = f_{d-1}$. The f-vector can be recovered from the h-vector, so both carry the same information. For a simplicial polytope of dimension $d$ one uses the f-vector and h-vector of its boundary complex, which has dimension $d - 1$.

Example: the boundary of an octahedron has $d = 3$ and $f = (1, 6, 12, 8)$. Then $(t-1)^3 + 6(t-1)^2 + 12(t-1) + 8 = t^3 + 3t^2 + 3t + 1$, so $h = (1, 3, 3, 1)$, and indeed $1 + 3 + 3 + 1 = 8 = f_2$.

Non-example of a nice property: the h-vector can have negative entries. For two disjoint edges, $d = 2$ and $f = (1, 4, 2)$, and $(t-1)^2 + 4(t-1) + 2 = t^2 + 2t - 1$ gives $h = (1, 2, -1)$.