Enumeration of Deutsch paths by adding the adding-a-new-slice method and applications

TL;DR

This paper introduces a novel enumeration method for a variation of Deutsch paths, utilizing the adding-a-new-slice technique and kernel method to count specific path features and subclasses, extending combinatorial enumeration techniques.

Contribution

It develops the adding-a-new-slice method for enumerating Deutsch paths with arbitrary down-step lengths and applies it to count maximal runs and specialized subclasses.

Findings

Derived formulas for counting Deutsch paths with arbitrary down-steps.

Extended enumeration techniques to path subclasses satisfying Stanley's conditions.

Demonstrated the effectiveness of the adding-a-new-slice method in path enumeration.

Abstract

A variation of Dyck paths allows for down-steps of arbitrary length, not just one. This is motivated by ideas due to Emeric Deutsch. We use the adding-a-new-slice technique and the kernel method to compute the number of maximal runs of up-step runs of length 1 and a subclass of Deutsch paths satisfying a condition that was stipulated by R. Stanley for Dyck paths.