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

  • Cyril Banderier
  • Michael WallnerEmail author
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 58)

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–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 \(y=x/\sqrt{2}\)), a problem that we study in a companion article.

Keywords

Lattice paths Generating function Analytic combinatorics Singularity analysis Kernel method Generalized Dyck paths Algebraic function Rational Catalan combinatorics Periodic support Bizley formula Grossman formula 

2010 Mathematics Subject Classification

Primary 05A15 Secondary 05A16 68W40 

Notes

Acknowledgements

This work is the result of a collaboration founded by the SFB project F50 “Algorithmic and Enumerative Combinatorics” and the Franco-Austrian PHC “Amadeus.” Michael Wallner is supported by the Austrian Science Fund (FWF) grant SFB F50-03 and by ÖAD, grant F04/2012. A preliminary version of this work [14] was presented at the conference ANALCO’15 (San Diego, January 2015) and at the 8th International Conference on Lattice Path Combinatorics and Applications (Pomona, August 2015). Last but not least, we thank Don Knuth, Ernst Schulte-Geers and Manuel Kauers for exchanging references on this problem, and the referee for the detailed feedback!

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Authors and Affiliations

  1. 1.Laboratoire d’Informatique de Paris-NordCNRS/Université Paris 13VilletaneuseFrance
  2. 2.Institute of Discrete Mathematics and GeometryTU WienViennaAustria
  3. 3.LaBRIUniversité de BordeauxBordeauxFrance

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