Related papers: Shellable flag simplicial complexes of non-simple …
We study the K\H{o}nig type property for non-simple polyominoes. We prove that, for closed path polyominoes, the polyomino ideals are of K\H{o}nig type, extending the results of Herzog and Hibi for simple thin polyominoes. As an application…
We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.
Let $\Phi$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $\Delta^m (\Phi)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $\Delta^m (\Phi)$ is…
We describe a natural geometric relationship between matroids and underlying flag matroids by relating the geometry of the greedy algorithm to monotone path polytopes. This perspective allows us to generalize the construction of underlying…
We recently introduced a notion of tilings of geometric realizations of finite relative simplicial complexes and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property of being shellable, a…
We classify the holomorphic parabolic geometries on compact complex manifolds of general type. We accomplish this by bounding the numerical dimension of any smooth projective variety in terms of geometric invariants of the flag variety…
We introduce a construction on a flag complex that, by means of modifying the associated graph, generates a new flag complex whose $h$-factor is the face vector of the original complex. This construction yields a vertex-decomposable, hence…
We relate properties of weighted flags (or multiflags) of type AD to statistics of the corresponding Weyl groups. For type A, we recover the Mahonian statistics on symmetric groups. Finally, we sketch briefly an easy extension incorporating…
A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Frechet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application we describe a class of coadjoint…
We find families of simplicial complexes where the simplicial chromatic polynomials defined by Cooper--de Silva--Sazdanovic \cite{CdSS} are Hilbert series of Stanley--Reisner rings of auxiliary simplicial complexes. As a result, such…
Let G be a simple undirected graph. We find the number of maximal independent sets in complete t-partite graphs. We will show that vertex decomposability and shellability are equivalent in this graphs. Also, we obtain an equivalent…
This paper expands on and refines some known and less well-known results about the finite subset spaces of a simplicial complex $X$ including their connectivity and their top homology groups. It also discusses the inclusion of the…
A flag manifold over a semifield K can be partitioned into "half i-circles" which are orbits of a K-action on that flag manifold. Here i is fixed and it corresponds to a simple reflection in the Weyl group. We prove (for certain K) a…
In this note we construct a flag simplicial $3$-sphere $\Delta$ with the following properties: - $\Delta$ is not a suspension; - $\Delta$ has no edge that can be contracted to obtain another flag sphere; - The only equators (induced…
We prove that for all $d \geq 1$ a shellable $d$-dimensional simplicial complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection…
We conjecture and prove closed-form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple algorithm which allows expression of any…
Grid polyominoes form a class of thin polyominoes with one or more holes arranged in a grid-like pattern in the plane. In this paper, we prove that the rook polynomial of grid polyominoes coincides with the h-polynomial of their…
Throughout this work, the vertex decomposability and shellability of graphs formed from other graphs by various operations are investigated. Also among the other things, by using some graph operations, new classes of Cohen-Macaulay graphs…
Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…
In this article, the comodule structure of Chow rings of Flag manifolds $\operatorname{CH}(G/B)$ is described by Schubert cells. Its equivariant version gives rise to a Hopf structure of the equivariant cohomology of flag manifolds…