Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we characterize the antichain cutsets in semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We ...

We study the set S nc(p, q) of annular non-crossing permutations of type B, and we introduce a corresponding set NC(p, q) of annular non-crossing partitions of type B, where p and q are two positive integers. We prove that the natural bijection between S nc(p, q) and NC (p, q) is a poset isomorphism, where the partial order on S nc(p, q) is induced from the hyperoctahedral group Bp+q, while NC ...

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra, self-dual with respect to a scalar product which counts the number of pictures (in the sense of Zelevinsky) between two do...

Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanne...

Let $\pi:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$ the image of standard basis. Motivated by Postnikov's study postitive Grassmannians via plabic graphs and Galashin's connection to slices zonotopal tilings 3-dimensional cyclic zonotopes, we poset subdivisions induced restriction $\pi$ $k$-th hypersimplex, for $k=1,\dots,n-1$. We show that: - For arbitrary $k\le d+1$, c...

Given an aane projection : P ! Q of a d-polytope P onto a polygon Q, it is proved that the poset of proper polytopal subdivisions of Q which are induced by has the homotopy type of a sphere of dimension d ? 3 if maps all vertices of P into the boundary of Q. This result, originally conjectured by Reiner, is an analogue of a result of Billera, Kapranov and Sturmfels on cellular strings on polyto...

Let Φ be an finite root system with corresponding reflection group W and let m be a nonnegative integer. We consider the generalized cluster complex ∆(Φ) defined by S. Fomin and N. Reading and the poset NC(m)(W ) of m-divisible noncrossing partitions defined by D. Armstrong. We give a characterization of the faces of ∆(Φ) in terms of NC(m)(W ), generalizing that of T. Brady and C. Watt given in...