A Second Look at the Toric H-polynomial of a Cubical Complex

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 GesselShapiro 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.