# 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} }

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

#### 53 Citations

Two non-holonomic lattice walks in the quarter plane

- Mathematics, Computer Science
- Theor. Comput. Sci.
- 2009

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

- Computer Science
- ISSAC
- 2021

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

- Mathematics
- 2008

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… Expand

Counting walks with large steps in an orthant

- Mathematics
- 2018

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… Expand

Infinite Orders and Non-D-finite Property of 3-Dimensional Lattice Walks

- Mathematics, Computer Science
- Electron. J. Comb.
- 2016

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

- Mathematics
- 2014

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 model… Expand

Classification of walks in wedges

- Mathematics
- 2007

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 restricted… Expand

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

- Mathematics, Computer Science
- Eur. J. Comb.
- 2017

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

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2016

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

- Mathematics, Computer Science
- Electron. J. Comb.
- 2015

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

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