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Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem

Authors Andrei Asinowski , Axel Bacher , Cyril Banderier , Bernhard Gittenberger



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Author Details

Andrei Asinowski
  • Technische Universität Wien, Institut für Diskrete Mathematik und Geometrie, https://dmg.tuwien.ac.at/asinowski/
Axel Bacher
  • Université Paris 13, LIPN, UMR CNRS 7030, http://lipn.univ-paris13.fr/~bacher
Cyril Banderier
  • Université Paris 13, LIPN, UMR CNRS 7030, http://lipn.univ-paris13.fr/~banderier
Bernhard Gittenberger
  • Technische Universität Wien, Institut für Diskrete Mathematik und Geometrie, https://dmg.tuwien.ac.at/gittenberger/

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Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger. Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 10:1-10:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.AofA.2018.10

Abstract

In a companion article dedicated to the enumeration aspects, we showed how to obtain closed form formulas for the generating functions of walks, bridges, meanders, and excursions avoiding any fixed word (a pattern p). The autocorrelation polynomial of this forbidden pattern p (as introduced by Guibas and Odlyzko in 1981, in the context of regular expressions) plays a crucial role. In this article, we get the asymptotics of these walks. We also introduce a trivariate generating function (length, final altitude, number of occurrences of p), for which we derive a closed form. We prove that the number of occurrences of p is normally distributed: This is what Flajolet and Sedgewick call an instance of Borges's theorem. We thus extend and refine the study by Banderier and Flajolet in 2002 on lattice paths, and we unify several dozens of articles which investigated patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. Our approach relies on methods of analytic combinatorics, and on a matricial generalization of the kernel method. The situation is much more involved than in the Banderier-Flajolet work: forbidden patterns lead to a wider zoology of asymptotic behaviours, and we classify them according to the geometry of a Newton polygon associated with these constrained walks, and we analyse what are the universal phenomena common to all these models of lattice paths avoiding a pattern.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Distribution functions
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Grammars and context-free languages
Keywords
  • Lattice paths
  • pattern avoidance
  • finite automata
  • context-free languages
  • autocorrelation
  • generating function
  • kernel method
  • asymptotic analysis
  • Gaussian limit law

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