A wide range of articles dealing with the occurrence of strings in Dyck paths appear frequently in the literature [2], [3], [5], [6], [7] and [11]. A Dyck path of semilength n is a lattice path of N2 running from (0, 0) to (2n, 0), whose allowed steps are the up diagonal step (1, 1) and the down diagonal step (1,−1). These steps are called rise and fall respectively. It is clear that each Dyck ...

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 certai...

We introduce a new lattice structure on Dyck words. We exhibit efficient algorithms to compute meets and joins of Dyck words.

The symmetric q, t-Catalan polynomial Cn(q, t), which specializes to the Catalan polynomial Cn(q) when t = 1, was defined by Garsia and Haiman in 1994. In 2000, Garsia and Haglund proved the existence of statistics a(π) and b(π) on Dyck paths such that Cn(q, t) = P π qt where the sum is over all n × n Dyck paths. Specializing t = 1 gives Cn(q) = P π q and specializing q = 1 as well gives the us...

Dyck paths are directed walks on Z starting at (0, 0) and ending on the line y = 0, which have no vertices with negative y-coordinates, and which have steps in the (1, 1) and (1,−1) directions [11]. We impose the additional geometrical constraint that the paths have height at most h, that is, they lie between lines y = 0 and y = h. Given a Dyck path π, we define the length n(π) to be half the n...

It is a well-known theorem by Chomsky and Schützenberger (1963) that every context-free language can be represented as a homomorphic image of the intersection of a Dyck language and a regular language. This paper gives a Chomsky-Schützenberger-type characterization for multiple context-free languages, which are a natural extension of context-free languages, with introducing the notion of multip...

Abstract. We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length n) to the case of p– watermelons with a wall (i.e., to a certain family of p nonintersecting Dyck paths; simple Dyck paths being the special case p = 1.) We work out this asymptotics for the case p = 2 only, since the computations involved are already qui...

In the present paper we consider the statistic “number of udu’s” in Dyck paths. The enumeration of Dyck paths according to semilength and various other parameters has been studied in several papers. However, the statistic “number of udu’s” has been considered only recently. Let Dn denote the set of Dyck paths of semilength n and let Tn, k, Ln, k, Hn, k and W (r) n, k denote the number of Dyck p...

We introduce the Dyck path triangulation of the cartesian product of two simplices ∆n−1×∆n−1. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of ∆rn−1 × ∆n−1 using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of t...