8 research outputs found
Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges\u27s Theorem
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
Tesis Doctoral inédita leÃda en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 15-09-201
Generalized Dyck paths of bounded height
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
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
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 ..