The toric h-vector of a cubical complex in terms of noncrossing partition statistics

TL;DR

This paper presents a new combinatorial statistic on noncrossing partitions that directly relates to the toric h-vector of cubical complexes, providing a simple interpretation and revealing symmetries via noncrossing partition lattice involutions.

Contribution

It introduces a novel statistic on noncrossing partitions that expresses the toric h-vector of cubical complexes in a new, combinatorial way, linking lattice involutions to geometric symmetries.

Findings

Expresses toric h-vector coordinates as total weights of noncrossing partitions

Provides a simple combinatorial interpretation of cubical shelling contributions

Derives symmetry properties from noncrossing partition lattice involutions

Abstract

This paper introduces a new and simple statistic on noncrossing partitions that expresses each coordinate of the toric $h$-vector of a cubical complex, written in the basis of the Adin $h$-vector entries, as the total weight of all noncrossing partitions. The same model may also be used to obtain a very simple combinatorial interpretation of the contribution of a cubical shelling component to the toric $h$-vector. In this model, a strengthening of the symmetry expressed by the Dehn-Sommerville equations may be derived from the self-duality of the noncrossing partition lattice, exhibited by the involution of Simion and Ullman.