S-Motzkin paths with catastrophes and air pockets

TL;DR

This paper introduces a novel combinatorial model called S-Motzkin paths that incorporate catastrophes and air pockets, analyzed through bivariate generating functions and the kernel method.

Contribution

It extends Motzkin path enumeration by integrating catastrophes and air pockets, providing new generating function techniques for their analysis.

Findings

Derived explicit bivariate generating functions.

Extended kernel method application for complex path structures.

Enhanced understanding of path enumeration with additional features.

Abstract

So called $S$-Motzkin paths are combined the concepts `catastrophes' and `air pockets. The enumeration is done by properly set up bivariate generating functions which can be extended using the kernel method.