Motivated by the theory of operads, we introduce new combinatorial objects, called shrubs, that generalize forests of rooted trees. We show that the species of shrubs is isomorphic to the species of series-parallel posets. Résumé. Motivé par des considérations sur les opérades, on introduit de nouveaux objets combinatoires, appelés arbustes, qui généralisent les forêts d’arbres enracinés. On mo...

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a “strong” way, is proved. The result is motivated by a conjecture of Graham’s concerning probability correlation inequalities...

The main objects of the present paper are (i) Hibi rings (toric arising from order polytopes posets), (ii) stable set perfect graphs), and (iii) edge graphs satisfying odd cycle condition). goal is to analyze those three toric discuss their structures in case where class groups have small rank. We prove that (i), torsionfree. More precisely, we give descriptions groups. Moreover, characterize p...

An affine projection π : P p → Qq of convex polytopes induces an inclusion map of the face posets i : F(Q) → F(P ). We define an order-preserving map of posets h : F(P ) → Suspp−qF(Q) such that for any filter J of Suspp−qF(Q), the map h restricts to a homotopy equivalence between the order complexes of h−1(J) and J . As applications we prove (1) A conjecture of Stanley [11] concerning the relat...

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w into w lgw chains. They also observed that the problem of on-line chain partitioning of general posets of width w could be reduced to First-Fit chain partitioning of (2w + 1)-ladder-free posets of width w, where an m-ladder is the transitive closure of the union of two incomparable chains x1 6 . . . 6...

A finite poset is an angle order if its points can be mapped into angular regions in the plane so that x precedes y in the poset precisely when the region for x is properly included in the region for y. We show that all posets of dimension four or less are angle orders, all interval orders are angle orders, and that some angle orders must have an angular region less than 180” (or more than 180”...

We study weight posets of weight multiplicity free (WMF) representations of reductive Lie algebras. Specifically, we are interested in relations between dimR and the number of edges in the Hasse diagram of the corresponding weight poset #E(R). We compute the number of edges and upper covering polynomials for the weight posets of all WMFrepresentations. We also point out non-trivial isomorphisms...

We consider the poset P (N ;A1, A2, . . . , Am) consisting of all subsets of a finite set N which do not contain any of the Ai’s, where the Ai’s are mutually disjoint subsets of N . The elements of P are ordered by inclusion. We show that P belongs to the class of Macaulay posets, i.e. we show a Kruskal-Katona type theorem for P . For the case that the Ai’s form a partition of N , the dual P ∗ ...

The linear span of isomorphism classes of posets, P, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space P to the algebra of polynomials in the non-commutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index of the face lattice of a convex polytope changes when taking the pyramid and the pri...

There are two central themes to research involving applications of probabilistic methods to partially ordered sets. The first of these can be described as the study of random partially ordered sets. Among the specific models which have been studied are: random labelled posets; random t-dimensional posets; and the transitive closure of random graphs. A second theme concentrates on the adaptation...