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 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.     

Matrices of Formal Power Series Associated to Binomial Posets

We introduce an operation that assigns to each binomial poset a partially ordered set for which the number of saturated chains in any interval is a function of two parameters. We develop a corresponding theory of generating functions involving noncommutative formal power series modulo the closure of a principal ideal, which may be faithfully represented by the limit of an infinite sequence of lower triangular matrix representations. The framework allows us to construct matrices of formal power series whose inverse may be easily calculated using the relation between the Möbius and zeta functions, and to find a unified model for the Tchebyshev polynomials of the first kind and for the derivative polynomials used to express the derivatives of the secant function as a polynomial of the tangent function.On leave from the Rényi Mathematical Institute of the Hungarian Academy of Sciences.     

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem–Mahler–Lech theorem, the ab-series of a level poset is shown to be a rational generating function in the non-commutative variables a and b. In the case the poset is also Eulerian, the analogous result holds for the cd-series. Using coalgebraic techniques a method is developed to recognize the cd-series matrix of a level Eulerian poset.     

Tchebyshev triangulations of stable simplicial complexes

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F-polynomials j_∑fj−1(j(x−1)/2), the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev transform of the second kind. Using the reverse implication of a theorem by Schelin we observe that the Tchebyshev transforms of Schur stable polynomials with real coefficients have interlaced real roots in the interval (−1,1), and present ways to construct simplicial complexes with Schur stable F-polynomials. We show that the order complex of a Boolean algebra is Schur stable. Using and expanding the recently discovered relation between the derivative polynomials for tangent and secant and the Tchebyshev polynomials we prove that the roots of the corresponding pairs of derivative polynomials are all pure imaginary, of modulus at most one, and interlaced.     

Enumeration by kernel positions for strongly Bernoulli type truncation games on words

We find the winning strategy for a class of truncation games played on words. As a consequence of the present author's recent results on some of these games we obtain new formulas for Bernoulli numbers and polynomials of the second kind and a new combinatorial model for the number of connected permutations of given rank. For connected permutations, the decomposition used to find the winning strategy is shown to be bijectively equivalent to King's decomposition, used to recursively generate a transposition Gray code of the connected permutations.     

On the Stanley ring of cubical complex

We investigate the properties of the Stanley ring of a cubical complex, a cubical analogue of the Stanley-Reisner ring of a simplicial complex. We compute its Hilbert series in terms of thef-vector, and prove that by taking the initial ideal of the defining relations, with respect to the reverse lexicographic order, we obtain the defining relations of the Stanley-Reisner ring of the triangulation via “pulling the vertices” of the cubical complex. Applying an old idea of Hochster we see that this ring is Cohen-Macaulay when the complex is shellable, and we show that its defining ideal is generated by quadrics when the complex is also a subcomplex of the boundary complex of a convex cubical polytope. We present a cubical analogue of balanced Cohen-Macaulay simplicial complexes: the class of edge-orientable shellable cubical complexes. Using Stanley's results about balanced Cohen-Macaulay simplicial complexes and the degree two homogeneous generating system of the defining ideal, we obtain an infinite set of examples for a conjecture of Eisenbud, Green, and Harris. This conjecture says that theh-vector of a polynomial ring inn variables modulo an ideal which has ann-element homogeneous system of parameters of degree two, is thef-vector of a simplicial complex.     

Generalizations of Eulerian partially ordered sets, flag numbers, and the Möbius function

A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Möbius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.ResumeUn ensemble partiellement ordonné est r-épais si chacun de ses intervalles ouverts non-vides contient au moins r éléments. Dans cet article nous étudions les vecteurs drapeaux des ensembles partiellement ordonnés gradués r-épais. Nous démontrons que le cône le plus petit contenant ces vecteurs est isomorphe au cône des vecteurs drapeaux des ensembles partiellement ordonnés gradués quelconques. Nous définissons aussi un k-analogue de la fonction de Möbius et des ensembles partiellement ordonnés k-eulériens qui sont 2k-épais. Nous caractérisons les ensembles partiellement ordonnés k-eulériens de plusieurs manières, et généralisons les équations de Dehn-Sommerville pour le vecteur drapeaux d'un ensemble partiellement ordonné k-eulérien. Nous démontrons une nouvelle inégalité optimale pour les ensembles partiellement ordonnés eulériens de rang 8.