Redundant generating functions in lattice path enumeration

TL;DR

This paper explores the use of redundant generating functions to analyze lattice path problems, revealing unexpected connections to Catalan numbers and proving a conjecture related to path enumeration.

Contribution

It introduces the application of redundant generating functions to lattice path enumeration, uncovering new relationships and proving a conjecture in the field.

Findings

Catalan numbers appear in paths below y=2x

Redundant generating functions simplify path counting

Proved a conjecture by Niederhausen and Sullivan

Abstract

A redundant generating function is a generating function having terms which are not part of the solution of the original problem. We use redundant generating functions to study two path problems. In the first application we explain a surprising occurrence of Catalan numbers in counting paths that stay below the line y = 2x. In the second application we prove a conjecture of Niederhausen and Sullivan.