Ioannis Tasoulas

Enumeration of strings in Dyck paths: A bijective approach

The statistics concerning the number of appearances of a string @t in Dyck paths as well as its appearances in odd and even level have been studied extensively by several authors using mostly algebraic methods. In this work a different, bijective approach is followed giving some known as well as some new results.

Equivalence classes of ballot paths modulo strings of length 2 and 3

Two paths are equivalent modulo a given string

Ordered trees and the inorder traversal

\tau

Nonleft peaks in Dyck paths

, whenever they have the same length and the positions of the occurrences of

The average number of times a stack exceeds a certain size

\tau

Counting strings in Dyck paths

are the same in both paths. This equivalence relation was introduced for Dyck paths in \cite{BP}, where the number of equivalence classes was evaluated for any string of length 2. In this paper, we evaluate the number of equivalence classes in the set of ballot paths for any string of length 2 and 3, as well as in the set of Dyck paths for any string of length 3.

Some strings in Dyck paths.

In this paper ordered trees are studied with respect to the inorder traversal. New decompositions of ordered trees are introduced and used to obtain enumeration results according to various parameters. Furthermore, the set of all ordered trees with prescribed degree sequence according to the inorder is studied with the aid of Dyck paths. This set is constructed and its cardinal number is evaluated recursively.

On the dominance partial ordering of Dyck paths

A peak in a Dyck path is called nonleft, if the ascent preceding it is greater than or equal to the descent following it. In this paper, we present a combinatorial construction of the set of Dyck paths of fixed semilength and number of nonleft peaks. As a bonus, we obtain various results on the enumeration of several kinds of peaks.

Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property

We obtain an explicit as well as an asymptotic formula for the average number of times a stack exceeds a fixed size j, after the execution of n push operations. The stack is assumed to have limitless capacity and all possible sequences of push and pop operations are considered to be equally likely.