compute f-and h-vector of simplicial complex

Question

A simplicial complex $\Delta$ is uniquely determined by its facets $\mathcal{F}(\Delta)$. I know how to compute the $f$-vector and therefore the $h$-vector given all the faces of $\Delta$. Now given the set of facets $\mathcal{F}(\Delta)$, can I compute the $f$-and $h$-vector of $\Delta$? If so, which program can do this?

Answer 1

You can write a fairly generic formula for the $f$-polynomial. It follows directly from inclusion-exclusion so I skip the proof:

$$ f_\Delta(t) = \sum_{\emptyset\neq S\subseteq \mathcal{F(\Delta)}} (-1)^{|S|+1}(t+1)^{|\bigcap S|}. $$

Every piece of software that allows you to construct a simplicial complex from facets will also compute its $f$-vector. Surely SAGE, Polymake, GAP can do it.