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

TL;DR

This paper provides a combinatorial proof for a formula calculating Betti numbers of the Stanley-Reisner ring associated with the boundary complex of stacked polytopes, connecting algebraic invariants with combinatorial structures.

Contribution

It offers the first combinatorial proof of a known algebraic formula for Betti numbers of stacked polytope boundary complexes.

Findings

Confirmed the formula for Betti numbers using combinatorial methods

Connected algebraic Betti numbers with polytope combinatorics

Extended understanding of Stanley-Reisner rings for stacked polytopes

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.