A second look at the toric h-polynomial of a cubical complex

TL;DR

This paper presents explicit formulas and combinatorial models for the toric h-polynomial of cubical complexes, establishing non-negativity and connections to noncrossing partitions and orthogonal polynomials.

Contribution

It introduces a new combinatorial model for toric h-polynomials, proving non-negativity and linking to classical polynomial families.

Findings

Explicit formula for toric h-contribution of cubical shelling components

New combinatorial model proving non-negativity of contributions

Connections to noncrossing partitions and orthogonal polynomials

Abstract

We provide an explicit formula for the toric $h$-contribution of each cubical shelling component, and a new combinatorial model to prove Clara Chan's result on the non-negativity of these contributions. Our model allows for a variant of the Gessel-Shapiro result on the $g$-polynomial of the cubical lattice, this variant may be shown by simple inclusion-exclusion. We establish an isomorphism between our model and Chan's model and provide a reinterpretation in terms of noncrossing partitions. By discovering another variant of the Gessel-Shapiro result in the work of Denise and Simion, we find evidence that the toric $h$-polynomials of cubes are related to the Morgan-Voyce polynomials via Viennot's combinatorial theory of orthogonal polynomials.