Dyck paths with catastrophes modulo the positions of a given pattern

TL;DR

This paper studies Dyck paths with catastrophes, classifying them based on the positions of small patterns, and provides generating functions and asymptotic formulas for their enumeration.

Contribution

It introduces a novel framework for analyzing Dyck paths with catastrophes through pattern-based equivalence classes and derives explicit generating functions.

Findings

Derived generating functions for p-equivalence classes

Provided asymptotic approximations for class counts

Established enumeration formulas for small pattern cases

Abstract

For any pattern $p$ of length at most two, we provide generating functions and asymptotic approximations for the number of $p$-equivalence classes of Dyck paths with catastrophes, where two paths of the same length are $p$-equivalent whenever the positions of the occurrences of the pattern $p$ are the same.