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

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

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