Relative blocking in posets

Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.Comment: 29 pages. Corrected version of original publication which is available at http://www.springerlink.com, see Corrigendu

Pattern Recognition on Oriented Matroids: Halfspaces, Convex Sets and Tope Committees

The principle of inclusion-exclusion is applied to subsets of maximal covectors contained in halfspaces of a simple oriented matroid and to convex subsets of its ground set for enumerating tope committees.Comment: 15 pages; v.2 - minor improvements, v.3,4 - new Section 7 and references adde

Generalizations of Eulerian partially ordered sets, flag numbers, and the Mobius 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 Mobius 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

A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings

Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on $f(n)$, the maximum number of stable matchings that a stable matching instance with $n$ men and $n$ women can have. It has been a long-standing open problem to understand the asymptotic behavior of $f(n)$ as $n\to\infty$, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately $2.28^n$, and the best upper bound was $2^{n\log n- O(n)}$. In this paper, we show that for all $n$, $f(n) \leq c^n$ for some universal constant $c$. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call "mixing". The latter might be of independent interest

The Polyhedral Geometry of Partially Ordered Sets

Pairs of polyhedra connected by a piecewise-linear bijection appear in different fields of mathematics. The model example of this situation are the order and chain polytopes introduced by Stanley in, whose defining inequalities are given by a finite partially ordered set. The two polytopes have different face lattices, but admit a volume and lattice point preserving piecewise-linear bijection called the transfer map. Other areas like representation theory and enumerative combinatorics provide more examples of pairs of polyhedra that are similar to order and chain polytopes. The goal of this thesis is to analyze this phenomenon and move towards a common theoretical framework describing these polyhedra and their piecewise-linear bijections. A first step in this direction was done by Ardila, Bliem and Salazar, where the authors generalize order and chain polytopes by replacing the defining data with a marked poset. These marked order and chain polytopes still admit a piecewise-linear transfer map and include the Gelfand-Tsetlin and Feigin-Fourier-Littelmann-Vinberg polytopes from representation theory among other examples. We consider more polyhedra associated to marked posets and obtain new results on their face structure and combinatorial interplay. Other examples found in the literature bear resemblance to these marked poset polyhedra but do not admit a description as such. This is our motivation to consider distributive polyhedra, which are characterized by describing networks by Felsner and Knauer analogous to the description of order polytopes by Hasse diagrams. For a subclass of distributive polyhedra we are able to construct a piecewise-linear bijection to another polyhedron related to chain polytopes. We give a description of this transfer map and the defining inequalities of the image in terms of the underlying network