On the Mobius function of a lower Eulerian Cohen-Macaulay poset

TL;DR

This paper establishes an inequality for the Mobius function of a specific class of posets, providing partial evidence for conjectures related to nonnegativity of certain combinatorial invariants in algebraic combinatorics.

Contribution

It introduces a new inequality for the Mobius function of a modified lower Eulerian Cohen-Macaulay poset, supporting conjectures on nonnegativity of toric and cubical h-vectors.

Findings

Proves an inequality for the Mobius function after adding a maximum element.

Supports Stanley's nonnegativity conjecture for toric h-vectors.

Provides evidence for Adin's nonnegativity conjecture for cubical h-vectors.

Abstract

A certain inequality is shown to hold for the values of the Mobius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen-Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley's nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen-Macaulay meet-semilattice and Adin's nonnegativity conjecture for the cubical h-vector of a Cohen-Macaulay cubical complex.