Lattice paths with infinitely many down steps -- the negative boundary model

TL;DR

This paper studies a variation of Dyck paths allowing multiple down-steps, providing enumeration methods using the kernel method and linear algebra, expanding understanding of lattice paths with complex step sets.

Contribution

It introduces a new class of lattice paths with multiple down-steps and develops enumeration techniques for paths confined in a strip.

Findings

Enumeration formulas for the new lattice paths

Application of kernel method and linear algebra techniques

Extension of classical Dyck path enumeration

Abstract

We consider a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ down-steps $(1,-j)$, for $j\ge2$ are allowed. We give credits to Emeric Deutsch for that. The enumeration of such objects living in a strip is performed. Methods are the kernel method and techniques from linear algebra.