We define the class of sofic-Dyck shifts which extends the class of Markov-Dyck shifts introduced by Inoue, Krieger and Mat-sumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta ...

The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0, 0) to (n, n) which is below the diagonal line y = x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0, 0) to (m, n) ∈ N2 which is below the diagonal line y...

A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an efficient scheme for encoding simple outerplanar maps. 1 Statement of results A map is a connected planar multigraph with a...

It is known that both the number of Dyck paths with 2n steps and k peaks, and the number of Dyck paths with 2n steps and k steps at odd height, follow the Narayana distribution. In this paper we present a bijection which explicitly illustrates this equinumeracy. Moreover, we extend this bijection to bilateral Dyck paths. The restriction to Dyck paths preserves the number of contacts.

Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length 2n and noncrossing partitions of [2n + 1] with n + 1 blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.

A new algorithms to generate all Dyck words is presented, which is used in ranking and unranking Dyck words. We emphasize the importance of using Dyck words in encoding objects related to Catalan numbers. As a consequence of formulas used in the ranking algorithm we can obtain a recursive formula for the nth Catalan number.

We show that the one-sided Dyck shift has a unique tail invariant topologically σ-finite measure (up to scaling). This invariant measure of the one sided Dyck turns out to be a shift-invariant probability. Furthermore, it is one of the two ergodic probabilities obtaining maximal entropy. For the two sided Dyck shift we show that there are exactly three ergodic double-tail invariant probabilitie...

In this paper configurations of n non-intersecting lattice paths which begin and end on the line y = 0 and are excluded from the region below this line are considered. Such configurations are called Hankel n−paths and their contact polynomial is defined by ẐH 2r(n;κ) ≡ ∑r+1 c=1 |H (n) 2r (c)|κc where H (n) 2r (c) is the set of Hankel n-paths which make c intersections with the line y = 0 the lo...

We study the relationship between rational slope Dyck paths and invariant subsets of Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)–Dyck paths and d-tuples of (n,m)-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and t...