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

TL;DR

This paper studies how ballot and Dyck paths can be grouped into equivalence classes based on the positions of specific substrings, extending previous work from length 2 to length 3.

Contribution

It generalizes the enumeration of equivalence classes of ballot and Dyck paths for strings of length 2 and 3, providing new combinatorial results.

Findings

Number of equivalence classes for ballot paths with substrings of length 2 and 3

Number of equivalence classes for Dyck paths with substrings of length 3

Extension of previous results from length 2 to length 3

Abstract

Two paths are equivalent modulo a given string $\tau$, whenever they have the same length and the positions of the occurrences of $\tau$ 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.