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Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects

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Language and Automata Theory and Applications (LATA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10792))

Abstract

This article presents a powerful method for the enumeration of pattern-avoiding words generated by an automaton or a context-free grammar. It relies on methods of analytic combinatorics, and on a matricial generalization of the kernel method. Due to classical bijections, this also gives the generating functions of many other structures avoiding a pattern (e.g., trees, integer compositions, some permutations, directed lattice paths, and more generally words generated by a push-down automaton). We focus on the important class of languages encoding lattice paths, sometimes called generalized Dyck paths. We 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 words. Indeed, we obtain formulas for the generating functions of walks/bridges/meanders/excursions avoiding any fixed word (a pattern). We show that the autocorrelation polynomial of this forbidden pattern (as introduced by Guibas and Odlyzko in 1981, in the context of regular expressions) still plays a crucial role for our algebraic functions. We identify a subclass of patterns for which the formulas have a neat form. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Our approach also opens the door to establish the universal asymptotics and limit laws for the occurrence of patterns in more general algebraic languages.

Work funded by the Austrian Science Fund (FWF) project SFB F50 “Algorithmic and Enumerative Combinatorics”.

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Notes

  1. 1.

    Some weights (or probabilities, or multiplicities) could be associated with each jump, but we omit them in this article to keep readability. All the proofs would be similar.

  2. 2.

    A similar notion also appears in the work of Schützenberger on synchronizing words [34].

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Authors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, 1040, Vienna, Austria

    Andrei Asinowski & Bernhard Gittenberger

  2. Laboratoire d’Informatique de Paris Nord, Université Paris 13, 99, rue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Axel Bacher & Cyril Banderier

Authors
  1. Andrei Asinowski
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  2. Axel Bacher
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  3. Cyril Banderier
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  4. Bernhard Gittenberger
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Corresponding authors

Correspondence to Andrei Asinowski , Axel Bacher , Cyril Banderier or Bernhard Gittenberger .

Editor information

Editors and Affiliations

  1. Bar-Ilan University, Ramat Gan, Israel

    Shmuel Tomi Klein

  2. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  3. Ariel University, Ariel, Israel

    Dana Shapira

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Asinowski, A., Bacher, A., Banderier, C., Gittenberger, B. (2018). Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects. In: Klein, S., Martín-Vide, C., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2018. Lecture Notes in Computer Science(), vol 10792. Springer, Cham. https://doi.org/10.1007/978-3-319-77313-1_15

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  • DOI: https://doi.org/10.1007/978-3-319-77313-1_15

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