Enumeration of Łukasiewicz paths modulo some patterns

Baril, Jean-Luc; Kirgizov, Sergey; Petrossian, Armen

doi:10.1016/j.disc.2018.12.005

Cited by 9 publications

(8 citation statements)

References 21 publications

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“…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. [5,16,25,33,38,53,57,60,62,66,69] and all the examples mentioned in our Sect. 8.…”

Section: Introductionmentioning

confidence: 76%

“…• Schröder bridges (also called Delannoy paths) with k occurrences of DD that cross y = 0 (see A110121 and [13,80]), • excursions from (0, 0) to (3n, 0) that use steps u = (2, 1), U = (1, 2), and D = (1, −1), and have k peaks uD or UD (see A108425 and [30]), • Łukasiewicz paths with k U-steps that start at an even level (see A091894 and [16,20]).…”

Section: Example 85 (Partially Directed Self-avoiding Walks)mentioning

confidence: 99%

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

et al. 2019

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In this article we develop a vectorial kernel method-a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. 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. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges's theorem. Keywords Lattice paths • Dyck paths • Motzkin paths • Łukasiewicz paths • Pattern avoidance • Autocorrelation • Finite automata • Markov chains • Pushdown automata • Generating functions • Wiener-Hopf factorization • Kernel method • Asymptotic analysis • Gaussian limit law • Borges' theorem We dedicate this article to the memory of Philippe Flajolet, our cheerful and inspiring mentor, founder of analytic combinatorics.

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Section: Introductionmentioning

confidence: 76%

Section: Example 85 (Partially Directed Self-avoiding Walks)mentioning

confidence: 99%

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

et al. 2019

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“…Now we present a graph (automaton) to recognize exactly the S-Motzkin paths: This subfamily of Motzkin paths originated from a question in a student competition; see [10] and [7] for history and analysis. In the following we will combine this family with catastrophes and air pockets, both originating in papers by Jean-Luc Baril and his team [5], [3]; the older paper by Banderier and Wallner [2] might be called the standard reference for lattice paths with catastrophes. The very recent papers [4,5] contain some bijective aspects.…”

Section: Introductionmentioning

confidence: 99%

S-Motzkin paths with catastrophes and air pockets

Prodinger¹

2023

Preprint

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“…On the other hand, in [2][3][4][5][6]10] the authors investigate equivalence relations on the sets of Dyck paths, Motzkin paths, skew Dyck paths, Lukasiewicz paths, and Ballot paths where two paths of the same length are equivalent whenever they coincide on all occurrences of a given pattern. The main goal of this study consists in extending these studies for Dyck paths with catastrophes by considering the analogous equivalence relation on E.…”

Section: Introduction and Notationmentioning

confidence: 99%