Is there anyone who can help me with this problem? Any hint to the solution would be appreciated!
Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. Show that the h- and f-vectors of $\Delta$ satisfy the following identity $\sum_{i=0}^{d}h_it^i(1+t)^{(d-i)}=\sum_{i=0}^{d}f_{i-1}t^i$.
Let $[n] = \{1, . . . , n\}$ be the vertex set and $Δ$ a simplicial complex on $[n].$ Thus $Δ$ is a collection of subsets of $[n]$ such that if $F \in Δ$ and $F' ⊂ F$, then F' $\in Δ$. Let $f_i = f_i(Δ)$ denote the number of faces of $Δ$ of dimension $i$. Thus in particular $f_0 = n$, if ${i} \in Δ$ for all $i \in [n]$. The sequence $f(Δ) = (f_0, f_1, . . . , f_{d−1})$ is called the f-vector of $Δ.$ Letting $f_{−1} = 1$, we define the $h$-vector $h(Δ) = (h_0, h_1, . . . , h_d)$ of $Δ $ by the formula $$\sum_{i=0}^df_{i−1}(t − 1)^{d−i} =\sum_{i=0}^dh_it^{d−i}.$$
