Skew Dyck paths with catastrophes

TL;DR

This paper studies a new class of lattice paths called skew Dyck paths with catastrophes, combining features of skew Dyck paths and paths with catastrophes, analyzed through generating functions and the kernel method.

Contribution

It introduces and analyzes skew Dyck paths with catastrophes, a novel combination of existing path models, using generating functions and the kernel method.

Findings

Derived generating functions for skew Dyck paths with catastrophes.

Applied the kernel method to analyze path enumeration.

Provided explicit formulas for path counts.

Abstract

Skew Dyck paths are like Dyck paths, but an additional south-west step $(-1,-1)$ is allowed, provided that the path does not intersect itself. Lattice paths with catastrophes can drop from any level to the origin in just one step. We combine these two ideas. The analysis is strictly based on generating functions, and the kernel method is used.