Lattice paths of slope 2/5

TL;DR

This paper studies the enumeration and asymptotic behavior of lattice paths with slope 2/5, providing new algebraic generating functions and identities, and extending results to other slopes like 2/3.

Contribution

It introduces a novel algebraic approach to analyze Dyck paths of slope 2/5 and generalizes the kernel method for other slopes, answering Knuth's problem.

Findings

Derived algebraic generating functions for slope 2/5 paths

Established new combinatorial identities related to lattice paths

Extended analysis to paths with slopes like 2/3

Abstract

We analyze some enumerative and asymptotic properties of Dyck paths under a line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014.Our approach relies on the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths. A key ingredient in the proof is the generalization of an old trick of Knuth himself (for enumerating permutations sortable by a stack),promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic,and they offer some new combinatorial identities, which can be also tackled in the A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar results for other slopes than 2/5, an interesting case being e.g. Dyck paths below the slope 2/3, which corresponds to the so called Duchon's club model.