Asymptotics of lattice walks via analytic combinatorics in several variables

@article{Melczer2020AsymptoticsOL,
  title={Asymptotics of lattice walks via analytic combinatorics in several variables},
  author={Stephen Melczer and Mark C. Wilson},
  journal={Discrete Mathematics \& Theoretical Computer Science},
  year={2020}
}
International audience We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential… 

Tables from this paper

Analytic Combinatorics in Several Variables : Effective Asymptotics and Lattice Path Enumeration. (Combinatoire analytique en plusieurs variables : asymptotique efficace et énumération de chemin de treillis)
TLDR
This thesis gives several new applications of ACSV to the enumeration of lattice walks restricted to certain regions, developing rigorous algorithms and giving the firstcomplexity results in this area under conditions which are broadly satisfied.
Lattice walks at the Interface of Algebra, Analysis and Combinatorics
Lattice paths are a classic object of mathematics, with applications in a wide range of areas including combinatorics, theoretical computer science and queuing theory. In the past ten years, several
Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior
TLDR
The analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions with non-smooth singular sets.
Weighted lattice walks and universality classes
Algorithmic Approaches for Lattice Path Combinatorics
TLDR
This tutorial will present the universe of lattice path classes, and survey some of the strategies used to deduce enumerative formulas, and discuss how to express the generating functions as diagonals of multivariate rational functions.
Computer Algebra in the Service of Enumerative Combinatorics
TLDR
An overview of recent results on structural properties and explicit formulas for generating functions of walks with small steps in the quarter plane are given, especially two important paradigms: "guess-and-prove" and "creative telescoping".
Counting walks with large steps in an orthant
In the past fifteen years, the enumeration of lattice walks with steps taken in a prescribed set S and confined to a given cone, especially the first quadrant of the plane, has been intensely
Computer Algebra for Lattice path Combinatorics
Classifying lattice walks in restricted lattices is an important problem in enumerative combinatorics. Recently, computer algebra has been used to explore and to solve a number of difficult questions
On the Nature of Four Models of Symmetric Walks Avoiding a Quadrant
We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite,
A combinatorial understanding of lattice path asymptotics
...
...

References

SHOWING 1-10 OF 41 REFERENCES
Analytic Combinatorics in Several Variables
This book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective. Analytic combinatorics is a branch of enumeration that uses analytic techniques to
Asymptotic Lattice Path Enumeration Using Diagonals
We consider d-dimensional lattice path models restricted to the first orthant whose defining step sets exhibit reflective symmetry across every axis. Given such a model, we provide explicit
Basic analytic combinatorics of directed lattice paths
Walks in the quarter plane: Kreweras’ algebraic model
We consider planar lattice walks that start from (0, 0), remain in the first quadrant i, j ≥ 0, and are made of three types of steps: North-East, West and South. These walks are known to have
Walks with small steps in the quarter plane
Let S be a subset of {-1,0,1}^2 not containing (0,0). We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadrant. A priori, there
An elementary solution of Gessel's walks in the quadrant
Classifying lattice walks restricted to the quarter plane
  • M. Mishna
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2009
Analytic Combinatorics
TLDR
This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.
Rational and algebraic series in combinatorial enumeration
Let A be a class of objects, equipped with an integer size such that for all n the number an of objects of size n is finite. We are interested in the case where the generating functionn antn is
...
...