The notion of exponential Dowling structures is introduced, generalizing Stanley’s original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley’s study of permutations associated with exponential structures l...

The notion of exponential Dowling structures is introduced, generalizing Stanley’s original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley’s study of permutations associated with exponential structures l...

The notion of exponential Dowling structures is introduced, generalizing Stanley’s original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley’s study of permutations associated with exponential structures l...

We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid what we call the semi-balanced case, that is, when each cycle has same number of edges pointing in two directions. Given a structure, identify natural class spanning trees show that, they induce shellable dissection polytope into maximal simplices. This allows for computation h ∗ $h^...

We consider the poset of weighted partitions Πn , introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Πn provide a generalization of the lattice Πn of partitions, which we show possesses many of the well-known properties of Πn. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal ...

Possibly the most fundamental combinatorial invariant associated to a finite simplicial complex is its f-vector, the integral sequence expressing the number of faces of the complex in each dimension. The h-vector of a complex is obtained by applying a simple invertible transformation to its f-vector, and thus the two contain the same information. Because some properties of the f-vector are easi...

We introduce the k-stellated (combinatorial) spheres and compare and contrast them with kstacked (triangulated) spheres. It is shown that for d ≥ 2k, any k-stellated sphere of dimension d bounds a unique and canonically defined k-stacked ball. In parallel, any k-stacked polytopal sphere of dimension d ≥ 2k bounds a unique and canonically defined k-stacked (polytopal) ball, which answers a quest...

Matrix Schubert varieties are the closures of orbits B×B acting on all n×n matrices, where B is group invertible lower triangular matrices. Extending work Fulton, Knutson and Miller identified a Gröbner basis for prime ideals these varieties. They also showed that corresponding initial Stanley-Reisner shellable simplicial complexes, derived related primary decomposition in terms reduced pipe dr...