An pa, bq-Dyck path P is a lattice path from p0, 0q to pb, aq that stays above the line y “ a b x. The zeta map is a curious rule that maps the set of pa, bq-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of P and zeta of P conjugate is enough to recover P . Our method begets an area-preserv...

We generalize the notion of pattern avoidance to arbitrary functions on ordered sets, and consider specifically three scenarios for permutations: linear, cyclic and hybrid, the first one corresponding to classical permutation avoidance. The cyclic modification allows for circular shifts in the entries. Using two bijections, both ascribable to both Deutsch and Krattenthaler independently, we sin...

We study the moments of orthogonal polynomial sequences (OPS) arising from tridiagonal matrices. We obtain combinatorial information about the sequence of moments of some OPS in terms of Motzkin and Dyck paths, and also in terms of the binomial transform. We then introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of...

Given a string σ over alphabet Σ and a grammar G defined over the same alphabet, how many minimum number of repairs: insertions, deletions and substitutions are required to map σ into a valid member of G ? We investigate this basic question in this paper for DYCK(s). DYCK(s) is a fundamental context free grammar representing the language of well-balanced parentheses with s different types of pa...

In [3] it was shown that the One sided Dyck is uniquely ergodic with respect to the one sided-tail relation, where the tail invariant probability is also shift invariant and obtains the topological entropy. In this paper we show that the two sided Dyck has a double-tail invariant probability, which is also shift invariant, with entropy strictly less than the topological entropy.

Stanley and Callan considered Dyck paths where the lengths of the run to the origin is always odd resp. the last one even, and the other ones odd. These subclasses are also enumerated by (shifted) Catalan numbers. We study the (average) height of these objects, assuming all such Dyck paths of length 2n to be equally likely, and find that it behaves like ∼ √ πn, as in the unrestricted case. This...

The XML community generally takes trees and hedges as the model for XML document instances and element content. In contrast, Berstel and Boasson have discussed XML documents in the framework of extended context-free grammar, modeling XML documents as Dyck strings and schemas as balanced grammars. How can these two models be brought closer together? We examine the close relatioship between Dyck ...

We introduce a normal form for context-free grammars, called Dyck normal form. This is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired nonterminals. This pairwise property allows to define a homomorphism from Dyck words to words generated by a grammar in Dyck normal form. We prove that for each context-fre...