The Kernel Method for Lattice Paths Below a Line of Rational Slope

Abstract

We analyze some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope 2 / 5. This answers Knuth’s problem #4 from his “Flajolet lecture” during the conference “Analysis of Algorithms” (AofA’2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities and has applications to a full class of problems involving some “periodicities.” A key ingredient in the proof is the generalization of an old trick by 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 also be tackled in the A = B spirit of Wilf–Zeilberger–Petkovsek. We show how to obtain similar results for any rational slope. An interesting case is, e.g., Dyck paths below the slope 2 / 3 (this corresponds to the so-called Duchon’s club model), for which we solve a conjecture related to the asymptotics of the area below such lattice paths. Our work also gives access to lattice paths below an irrational slope (e.g., Dyck paths below \(y=x/\sqrt{2}\)), a problem that we study in a companion article.