Dyck algebras, interval temporal logic and posets of intervals

We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, as well as of regular elements. We also find a logic-theoretic interpretation of such Heyting algebras, which we call Dyck algebras, by showing that they are the algebraic counterpart of a certain fragment of a classical interval temporal logic (also known as Halpern-Shoham logic). Finally, we propose a generalization of our approach, suggesting a similar study of the Heyting algebra arising from the poset of intervals of a finite poset using Birkh\"off duality. In order to illustrate this, we show how several combinatorial parameters of Dyck paths can be expressed in terms of the Heyting algebra structure of Dyck algebras together with a certain total order on the set of atoms of each Dyck algebra.Comment: 17 pages, 3 figure

Greedy algorithms and poset matroids

We generalize the matroid-theoretic approach to greedy algorithms to the setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli (1998) [BNP]. We illustrate our result by providing a generalization of Kruskal algorithm (which finds a minimum spanning subtree of a weighted graph) to abstract simplicial complexes

Enumeration of saturated chains in Dyck lattices

We determine a general formula to compute the number of saturated chains in Dyck lattices, and we apply it to find the number of saturated chains of length 2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2 and 3) and the cardinality of the underlying poset.Comment: 9 page

Catalan-like numbers and succession rules

The ECO method and the theory of Catalan-like numbers introduced by Aigner seems two completely unrelated combinatorial settings. In this work we try to establish a bridge between them, aiming at starting a (hopefully) fruitful study on their interactions. We show that, in a linear algebra context (more precisely, using infinite matrices), a succession rule can be translated into a (generalized) Aigner matrix by means of a suitable change of basis in the vector space of one-variable polynomials. We provide some examples to illustrate this fact and apply it to the study of two particular classes of succession rules.Comment: Submitted. The paper has been presented at the conference "Paths, Permutations and Trees", held in Tianjin, 2004, February, 25-2

The M\"obius function of the consecutive pattern poset

An occurrence of a consecutive permutation pattern $p$ in a permutation $\pi$ is a segment of consecutive letters of $\pi$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the M\"obius function of intervals in this poset, providing what may be called a complete solution to the problem. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the M\"obius function. In particular, we show that the M\"obius function only takes the values -1, 0 and 1.Comment: 10 pages, 2 figure