4,447 research outputs found
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 in a permutation
is a segment of consecutive letters of whose values appear in the same
order of size as the letters in . 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
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