Graded Betti numbers of cycle graphs and standard Young tableaux

TL;DR

This paper establishes a bijective proof linking the Betti numbers of the Stanley-Reisner ring of cycle graphs to the enumeration of standard Young tableaux, revealing a combinatorial interpretation of algebraic invariants.

Contribution

It provides the first bijective proof connecting Betti numbers of cycle graphs' Stanley-Reisner rings to standard Young tableaux counts.

Findings

Betti numbers correspond to counts of standard Young tableaux

Bijection between algebraic invariants and combinatorial objects

Enhanced understanding of the algebraic-combinatorial relationship

Abstract

We give a bijective proof that the Betti numbers of a minimal free resolution of the Stanley-Reisner ring of a cycle graph (viewed as a one-dimensional simplicial complex) are given by the number of standard Young tableaux of a given shape.