We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the Möbius function and topology of such pattern posets. We prove our results using a poset fibration based on the embeddings of the poset, where embeddings are representations of occurrences. We show that the Möbius ...

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form “is x < y?”. This application motivates looking for stronger notions of dimensi...

We present a method of computing the generating function fP (x) of P -partitions of a poset P . The idea is to introduce two kinds of transformations on posets and compute fP (x) by recursively applying these transformations. As an application, we consider the partially ordinal sum Pn of n copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the seque...

We show that plane bipolar posets (i.e., orientations with no transitive edge) and transversal structures can be set in correspondence to certain (weighted) models of quadrant walks, via suitable specializations a bijection due Kenyon, Miller, Sheffield Wilson. then derive exact asymptotic counting results, particular we prove the number $$t_n$$ on $$n+2$$ vertices satisfies (for some $$c>0$$ )...

Given a directed graph G = (V,E) and an integer k ≥ 1, a Steiner k-transitive-closure-spanner (Steiner k-TC-spanner) of G is a directed graph H = (VH , EH) such that (1) V ⊆ VH and (2) for all vertices v, u ∈ V , the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwise. Motivated by applications to property reconstruction and access control hierarchies, we conce...

Abstract It is elementary and well known that if an element x of a bounded modular lattice L $ \mathbf L has complement in then relative every interval [ , b ] containing . We show the relatively strong assumption modularity can be replaced by weaker one formulated language so-call...

We study r-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite pro...

We investigate avoidance in (2+2)-free partially ordered sets, posets that do not contain any induced subposet isomorphic to the union of two disjoint chains of length two. In particular, we are interested in enumerating the number of partially ordered sets of size n avoiding both 2+2 and some other poset α. For any α of size 3, the results are already well-known. However, out of the 15 such α ...

We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations defined in terms of a new type of pattern....

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) Merris (2002), threshold split form upward-closed sets within they belong to, i.e., any sequence majorizing a or must itself be threshold, respectively. Motivated fact that have characterizations in terms forbidden induced subgraphs, we define class $\ma...