There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the twodimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams ...

We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in previous literature in a solution of the stationary state of the ‘TASEP’ stochastic process.

In the first part of this article we present a realization of the m-Tamari lattice T (m) n in terms of m-tuples of Dyck paths of height n, equipped with componentwise rotation order. For that, we define the m-cover poset P〈m〉 of an arbitrary bounded poset P , and show that the smallest lattice completion of the m-cover poset of the Tamari lattice Tn is isomorphic to the m-Tamari lattice T (m) n...

In her dissertation, Angela Hicks showed that, to reduce the HMZ shuffle conjecture, it is sufficient to find a certain bijection between two sets of two-part parking functions which, roughly speaking, either swaps the two parts of the parking function or transfers a car from the second part to the first part and which satisfies certain nice properties. I will consider a special case where this...

Many program-analysis problems can be formulated as graph-reachability problems. Interleaved Dyck language reachability ( InterDyck -reachability) is a fundamental framework to express wide variety of over edge-labeled graphs. The represents an intersection multiple matched-parenthesis languages (i.e., languages). In practice, program analyses typically leverage one achieve context-sensitivity,...

The starting point of this work is the discovery of a new and direct construction that relies bijectively the permutations of length n to some weighted Dyck paths named subdivided Laguerre histories. These objects correspond to the combinatorial interpretation of the development of the generating function for factorial numbers in terms of a Stieltjes continued fraction [9]. Such a bijection has...

One of the most recent papers on patterns occurring k times in Dyck paths was written by A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, 2007, to appear in Discrete Mathematics [5]. The authors find generating functions for all 16 patterns generated by combinations of four up (ր) and down (ց) steps. A Dyck path starts at (0, 0), takes only up and down steps, and ends a...

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 prove that the multivariate zeta function of a sofic-Dyck shift is the commutative series of some visibly pushdown language. As a consequence the zeta function of a sofic-Dyck shift is the generating function of a visibly pushdown language and is thus an N-algebraic series.