The Betti numbers of forests

TL;DR

This paper derives a recursive combinatorial formula for Betti numbers of forest-associated Stanley-Reisner ideals, introducing a new invariant for forests and suggesting its extension to general graphs.

Contribution

It provides a recursive formula for Betti numbers of forest ideals and defines a new numerical invariant for forests based on projective dimension.

Findings

Recursive formula for Betti numbers of forest ideals

Introduction of a new numerical invariant for forests

Proposal to extend the invariant to general graphs

Abstract

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs.