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

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

ON THE STRUCTURE AND INVARIANTS OF CUBICAL COMPLEXES

This dissertation introduces two new results for cubical complexes. The first is a 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. This expression can then be used to obtain a simple combinatorial interpretation of the contribution of a cubical shelling component to the toric h-vector. Secondly, a class of indecomposable permutations, bijectively equivalent to stan- dard double occurrence words, may be used to encode one representative from each equivalence class of the shellings of the boundary of the hypercube. Finally, an adja- cent transposition Gray code is constructed for this class of permutations, which can be implemented in constant amortized time

Neighborly Cubical Polytopes and Spheres

We prove that the neighborly cubical polytopes studied by G"unter M. Ziegler and the first author arise as a special case of neighborly cubical spheres constructed by Babson, Billera, and Chan. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz, and Wills can be embedded into R^3.Comment: 17 pages, 13 figure