A combinatorial proof of a formula for Betti numbers of a stacked polytope

Abstract

For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}(k[\Delta])$ of the Stanley-Reisner ring $k[\Delta]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $\beta_{k-1,k}(k[\Delta])=(k-1)\binom{n-d}{k}$. We prove this combinatorially.