Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

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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“…• enumeration in Tamari-like posets (see e.g. A007477 and [5,17]), • a mathematical model for bobbin lace (see A291083 and [49]).…”

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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“…Also, the nth little Schröder number r n counts the number of Schröder paths of length 2n with no H-steps at X-axis [34, A001003]. Recently, several authors [13,14,15,19,20,41,42] have considered lattice paths with various steps, including vertical steps permitted. In this paper, we consider a kind of generalized Motzkin paths, called G-Motzkin paths for short.…”

Section: Introductionmentioning

confidence: 99%

Some statistics on generalized Motzkin paths with vertical steps

Sun¹,

Zhao²,

Shi³

et al. 2022

Preprint

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Recently, several authors have considered lattice paths with various steps, including vertical steps permitted. In this paper, we consider a kind of generalized Motzkin paths, called G-Motzkin paths for short, that is lattice paths from (0, 0) to (n, 0) in the first quadrant of the XOY -plane that consist of up steps u = (1, 1), down steps d = (1, −1), horizontal steps h = (1, 0) and vertical steps v = (0, −1). We mainly count the number of G-Motzkin paths of length n with given number of z-steps for z ∈ {u, h, v, d}, and enumerate the statistics "number of z-steps" at given level in G-Motzkin paths for z ∈ {u, h, v, d}, some explicit formulas and combinatorial identities are given by bijective and algebraic methods, some enumerative results are linked with Riordan arrays according to the structure decompositions of G-Motzkin paths. We also discuss the statistics "number of z 1 z 2 -steps" in G-Motzkin paths for z 1 , z 2 ∈ {u, h, v, d}, the exact counting formulas except for z 1 z 2 = dd are obtained by the Lagrange inversion formula and their generating functions.

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Lattice Path Enumeration, The Kernel Method, and Diagonals

Melczer

2020

Texts &Amp; Monographs in Symbolic Computation

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Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

Cited by 5 publications

References 8 publications

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

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

Some statistics on generalized Motzkin paths with vertical steps

Lattice Path Enumeration, The Kernel Method, and Diagonals

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