Symmetry and stabilization of the last row of the Betti diagram for weighted-hyperplane monomial ideals

Establish whether, for I ∈ I_n(α, β) with n ≥ 5 and β ≥ 2 such that β_{n−2, n + a_2(R/I)} is an extremal Betti number of R/I, the last row of the Betti diagram of R/I is symmetric; furthermore, determine whether this last row stabilizes as α > β varies while the graph G determining I is fixed.

Background

After deriving explicit formulas for extremal Betti numbers in various cases and giving a classification of pseudo-Gorenstein rings among the studied ideals, the authors raise a conjectural observation about structural features of the Betti diagrams for their class I ∈ I_n(α, β).

They note that computational evidence (via Macaulay2) suggests a symmetric and stabilizing pattern in the last row of the Betti diagram under the stated conditions, and present this as a conjectural direction for future verification and paper.

References

Finally, we end this article with a conjectural observation. Let I∈ I_n(α,β) for n ≥ 5. When β_{n-2, n+a_2}(R/I) is an extremal Betti number and β ≥ 2, our computations using Macaulay2 suggest that the last row of the Betti diagram of R/I would be symmetric; moreover, it should stabilize when the graph G is fixed and α > β ≥ 2 vary.

— Extremal Betti numbers of certain two-dimensional monomial ideals (2510.12149 - Loc et al., 14 Oct 2025) in Remark, Section 6 (Applications)