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

Abstract

We dedicate this article to the memory of Philippe Flajolet, who was and will remain a guide and a wonderful source of inspiration for so many of us. UUU AbstractWe analyse 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-Petkovšek.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/ √ 2), a problem that we study in a companion article.Lattice paths below a line of rational slope 1 The "kernel method" that we mention here for functional equations in combinatorics has nothing to do with what is known as the "kernel method" or "kernel trick" in statistics or machine learning. Also, there is no integral directly related to our kernel. For sure, in our case the word kernel was chosen as its zeros will play a key role, and also, in one sense, as this kernel has in its core the full description of the problem, and its resolution.

Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method

“…The smallest non-Łukasiewicz cases are the Duchon lattice paths (steps S = {−2, +3}), and the Knuth lattice paths (steps S = {−2, +5}). Their enumerative and asymptotic properties are the subject of another article in this volume [11]. For these two families of lattice paths, the asymptotics are tricky, because the generating functions involve several dominant singularities.…”

“…The method was later generalized to enumeration and asymptotic analysis of directed lattice paths with any set of steps, and many other combinatorial structures enumerated by bivariate or trivariate functional equations (see, e.g., [6,8,19,20,26,27]). We refer to the introduction of [11] for a more detailed history of the kernel method.…”

Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

“…Our approach thus generalizes the enumeration and asymptotics obtained by Banderier and Flajolet [9] for classical lattice paths to lattice paths avoiding a given pattern. This work continues the tradition of investigation of enumerative and asymptotic properties of lattice paths via analytic combinatorics [9,10,12,14]. This allows us to unify the considerations of many articles which investigated natural patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths, corresponding patterns in trees, compositions, …; see e.g.…”

“…[9,24,39,40]). We refer to [14] for more on the long history and the numerous evolutions of this method, which found many applications e.g. for planar maps, permutations, lattice paths, directed animals, polymers, may it be in combinatorics, statistical mechanics, computer algebra, or in probability theory.…”

“…For Property 2, it is proven in[7] that several dominant singularities appear if and only if the gcd of the pairwise differences of the steps is not 1. In this case, the asymptotics are obtained via[14, Theorem 8.8]. Moreover, polar singularities are possible, but these are easy to handle.• For Property 3, it is satisfied in many natural cases (like e.g.…”