Ballot Paths Avoiding Depth Zero Patterns

TL;DR

This paper studies ballot paths that avoid specific patterns, extending previous work on Dyck paths by focusing on paths that stay above the diagonal and avoid certain subpatterns, providing new combinatorial insights.

Contribution

It generalizes the counting of pattern-avoiding paths from Dyck paths to ballot paths, offering new enumeration results and combinatorial characterizations.

Findings

Derived formulas for counting ballot paths avoiding given patterns

Extended pattern avoidance results from Dyck paths to ballot paths

Provided combinatorial interpretations for the avoidance counts

Abstract

In a paper by Sapounakis, Tasoulas, and Tsikouras \cite{stt}, the authors count the number of occurrences of patterns of length four in Dyck paths. In this paper we specify in one direction and generalize in another. We only count ballot paths that avoid a given pattern, where a ballot path stays weakly above the diagonal $y=x$, starts at the origin, and takes steps from the set $\{\uparrow ,\to \}=\{u,r\}$. A pattern is a finite string made from the same step set; it is also a path. Notice that a ballot path ending at a point along the diagonal is a Dyck path.