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 >= 3, then beta(k-1,k)(k[Delta])=(k-1)((n-d)(k)) We prove this combinatorially.