We make explicit a description in terms of convex geometry of the higher Bruhat orders B(n, d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher Stasheff-Tamari poset S1(n, d). We show that the map f sketched by Kapranov and Voevodsky from B(n, d) to S1([0, n + 1], d + 1) coincides with the map constructed by Rambau, and is a surjection for d ≤ 2. We also give g...

We introduce a partial order structure on the set of interval orders of a given size, and prove that such a structure is in fact a lattice. We also provide a way to compute meet and join inside this lattice. Finally, we show that, if we restrict to series parallel interval order, what we obtain is the classical Tamari poset.

Over the past 15 years, Ramsey theoretic techniques and concepts have been applied with great success to partially ordered sets. In the last year alone, four new applications of Ramsey theory to posets have produced solutions to some challenging combinatorial problems. First, Kierstead and Trotter showed that dimension for interval orders can be characterized by a single ramsey trail by proving...

We show that a minimal triangulation of the associahedron (Stasheff polytope) of dimension n is made of (n+ 1)n−1 simplices. We construct a natural bijection with the set of parking functions from a new interpretation of parking functions in terms of shuffles. Introduction The Stasheff polytope, also known as the associahedron, is a polytope which comes naturally with a poset structure on the s...

We completely characterize the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n) = n! has...

In this paper, we present formulas for the number of decompositions of elements of the Weyl groups of type A., D. and B, as products of a number of reflections that is not necessarily minimal. For this purpose, we consider the poset of conjugacy classes of W introduced in B6dard and Goupil (1992) for the symmetric group. This poset describes the action of the set of reflections of a reflection ...

Interpretation (AI) is a theoretical framework introduced by Cousot and Cousot in [10] to manipulate abstractions of program states. An abstraction can be used to simplify program analysis problems otherwise not computable in realistic time, to manageable problems more easily solvable. Instead of working on the concrete semantics of a program1, AI computes results over an abstract semantics all...

Stanely, R.P., f-vectors and h-vectors of simplicial posets, Journal of Pure and Applied Algebra 71 (1991) 319-331. A simplicial poset is a (finite) poset P with d such that every interval [6, x] is a boolean algebra. Simplicial posets are generalizations of simplicial complexes. The f-vector f(P) = (f,, f,, , ,f_,) of a simplicial poset P of rank d is defined by f; = #{x E P: [6, x] g B,, I}, ...

We study a poset of compositions restricted by part size under a partial ordering introduced by Björner and Stanley. We show that our composition poset Cd+1 is isomorphic to the poset of words A ∗ d. This allows us to use techniques developed by Björner to study the Möbius function of Cd+1. We use counting arguments and shellability as avenues for proving that the Möbius function is μ(u,w) = (−...

Let P i n be the poset of partitions of an integer n, ordered by re-nement. Let b(P i n) be the largest size of a level and d(P i n) be the largest size of an antichain of P i n. We prove that d(P i n) b(P i n) e + o(1) as n ! 1: The denominator is determined asymptotically. In addition, we show that the incidence matrices in the lower half of P i n have full rank, and we prove a tight upper bo...