8 research outputs found

    Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges\u27s Theorem

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    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\u27s 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

    Algebra, Geometry and Topology of the Riordan Group

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    Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 15-09-201

    Generalized Dyck paths of bounded height

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    Generalized Dyck paths (or discrete excursions) are one-dimensional paths that take their steps in a given finite set S, start and end at height 0, and remain at a non-negative height. Bousquet-M\'elou showed that the generating function E_k of excursions of height at most k is of the form F_k/F_{k+1}, where the F_k are polynomials satisfying a linear recurrence relation. We give a combinatorial interpretation of the polynomials F_k and of their recurrence relation using a transfer matrix method. We then extend our method to enumerate discrete meanders (or paths that start at 0 and remain at a non-negative height, but may end anywhere). Finally, we study the particular case where the set S is symmetric and show that several simplifications occur

    Walks confined in a quadrant are not always D-finite

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    We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z^2, and always stay in the quadrant x >= 0, y >= 0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the ordinary square lattice walks. Then, we prove that walks that start from (1,1), take their steps in {(2,-1), (-1,2)} and stay in the first quadrant have a non-D-finite generating function. Our proof relies on a functional equation satisfied by this generating function, and on elementary complex analysis.Comment: To appear in Theoret. Comput. Sci. (special issue devoted to random generation of combinatorial objects and bijective combinatorics

    Underdiagonal Lattice Paths With Unrestricted Steps

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    We use some combinatorial methods to study underdiagonal paths (on the Z 2 lattice) made up of unrestricted steps, i.e., ordered pairs of non-negative integers. We introduce an algorithm which automatically produces some counting generating functions for a large class of these paths. We also give an example of how we use these functions to obtain some specific information on the number d n;k of paths from the origin to a generic point (n; n \Gamma k): Keywords: underdiagonal lattice paths, paths with unrestricted steps, context-free grammars, generating functions, Schutzenberger methodology. 1 Introduction In his interesting paper [3], Gessel gives an algebraic method he calls "factorization of formal Laurent series" to find the generating functions for underdiagonal lattice paths with unrestricted steps (functions f 0 and f \Gamma in his notation) by means of the bivariate generating function of all lattice Z 2 's paths. By unrestricted steps we mean ordered pairs (ffi; ffi 0 ..

    Subject Index Volumes 1–200

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    Subject index volumes 1–92

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