Classifying lattice walks restricted to the quarter plane

@article{Mishna2009ClassifyingLW,
  title={Classifying lattice walks restricted to the quarter plane},
  author={Marni Mishna},
  journal={J. Comb. Theory, Ser. A},
  year={2009},
  volume={116},
  pages={460-477}
}
  • M. Mishna
  • Published 2009
  • Mathematics, Computer Science
  • J. Comb. Theory, Ser. A
This work considers the nature of generating functions of random lattice walks restricted to the first quadrant. In particular, we find combinatorial criteria to decide if related series are algebraic, transcendental holonomic or otherwise. Complete results for walks taking their steps in a maximum of three directions of restricted amplitude are given, as is a well-supported conjecture for all walks with steps taken from a subset of {0,+/-1}^2. New enumerative results are presented for several… Expand

Figures, Tables, and Topics from this paper

Two non-holonomic lattice walks in the quarter plane
TLDR
The non-holonomicity is established using the iteratedkernel method, a variant of the kernel method, which adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. Expand
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". Expand
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, thereExpand
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 intenselyExpand
Infinite Orders and Non-D-finite Property of 3-Dimensional Lattice Walks
TLDR
This paper confirms the conjectures that many models associated to a group of order at least $200$ and conjectures these groups were in fact infinite groups, and considers the non-D-finite property of the generating function for some of these models. Expand
Variants of the Kernel Method for Lattice Path Models
The kernel method has proved to be an extremely versatile tool for exact and asymptotic enumeration. Recent applications in the study of lattice walks have linked combinatorial properties of a modelExpand
Classification of walks in wedges
Planar lattice walks are combinatorial objects which arise in statistical mechanics in both the modeling of polymers and percolation theory. It has been shown previously that lattice walks restrictedExpand
Hypergeometric expressions for generating functions of walks with small steps in the quarter plane
TLDR
The first proof that these equations are indeed satisfied by the corresponding generating functions is given, which proves that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. Expand
Square lattice walks avoiding a quadrant
TLDR
This work investigates the two most natural cases of enumeration of walks in non-convex cones, and obtains closed form expressions for the number of n-step walks ending at certain prescribed endpoints, as a sum of three hypergeometric terms. Expand
On the Number of Walks in a Triangular Domain
TLDR
The central result is an explicit formula for the generating function of walks starting at a fixed point in this domain and ending anywhere within the domain. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 35 REFERENCES
Two non-holonomic lattice walks in the quarter plane
TLDR
The non-holonomicity is established using the iteratedkernel method, a variant of the kernel method, which adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. Expand
Transcendence of Generating Functions of Walks on the Slit Plane
Consider a single walker on the slit plane, that is, the square grid Z2 without its negative x-axis, who starts at the origin and takes his steps from a given set 6. Mireille Bousquet-MelouExpand
Basic analytic combinatorics of directed lattice paths
TLDR
A unified enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes based on a specific “kernel method” that provides an important decomposition of the algebraic generating functions involved, as well as a generic study of singularities of an associated algebraic curve. Expand
Walks confined in a quadrant are not always D-finite
TLDR
It is proved 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. Expand
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 haveExpand
Walks on the slit plane
Abstract. In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line . We call them walks on the slit plane. WeExpand
Bijective counting of Kreweras walks and loopless triangulations
  • O. Bernardi
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 2007
TLDR
This work gives the first purely combinatorial explanation of Germain Kreweras' counting formula and obtains simultaneously a bijective way of counting loopless triangulations. Expand
Partially directed paths in a wedge
TLDR
This paper considers a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedgedefined by Y=+/-pX, and an asymmetric wedge defined by the lines Y=pX and Y=0, where p>0 is an integer. Expand
Walks on the Slit Plane: Other Approaches
TLDR
Two new approaches for solving slit plane models are presented: one is inspired by an argument of Lawler; it is more combinatorial, and explains the algebraicity of the product of three series related to the model, and works for any set of steps S. Expand
The Holonomic Ansatz I. Foundations and Applications to Lattice Path Counting
Abstract.Many combinatorial quantities belong to the holonomic ansatz. For example, sequences enumerating lattice paths. Once this fact is known, many times empirically obtained “conjectures” can beExpand
...
1
2
3
4
...