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.
In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not have the same intuitive probabilistic behaviour as classical Dyck paths (the typical properties of which are strongly related to Brownian motion theory), and this article quantifies some relations between these two types of paths. We give a bijection with some other lattice paths and a link with a continued fraction expansion. Furthermore, we prove several formulae for related combinatorial structures conjectured in the On-Line Encyclopedia of Integer Sequences. Thanks to the kernel method and via analytic combinatorics, we provide the enumeration and limit laws of these "lattice paths with catastrophes" for any finite set of jumps. We end with an algorithm to generate such lattice paths uniformly at random.
This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints such as, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps −h, . . . , −1, +1, . . . , +h. The case h = 1 is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like numbers are known. The case h = 2 corresponds to "basketball" walks, which we treat in full detail. Then we move on to the more general case of walks with any finite set of steps, also allowing some weights/probabilities associated with each step. We show how a method of wide applicability, the so-called "kernel method", leads to explicit formulas for the number of walks of length n, for any h, in terms of nested sums of binomials. We finally relate some special cases to other combinatorial problems, or to problems arising in queuing theory.
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