A combinatorial proof of a formula for Betti numbers of a stacked polytope (Q2380441)

Summary: For a simplicial complex \(\Delta\), the graded Betti number \(\beta_{i,j}({\mathbf k}[\Delta])\) of the Stanley-Reisner ring \({\mathbf k}[0]\) over a field \({\mathbf k}\) has a combinatorial interpretation due to Hochster. \textit{N. Terai} and \textit{T. Hibi} [Manuscr. Math. 92, No. 4, 447--453 (1997; Zbl 0882.13018)] showed that if \(\Delta\) is the boundary complex of a \(d\)-dimensional stacked polytope with \(n\) vertices for \(d\geq 3\), then \(\beta_{k-1,k}({\mathbf k}[\Delta])= (k-1){n-d\choose k}\). We prove this combinatorially.