Betti numbers of Stanley--Reisner rings with pure resolutions

Let $\mathrm{\Delta }$ be simplicial complex and let $k[\mathrm{\Delta }]$ denote the Stanley–Reisner ring corresponding to $\mathrm{\Delta }$. Suppose that $k[\mathrm{\Delta }]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert–Samuel multiplicity of $k[\mathrm{\Delta }]$ in terms of the $h$–vector of $\mathrm{\Delta }$. As an application, we derive a linear equation system for the components of the $h$–vector of the clique complex of an arbitrary chordal graph.