相关论文: Enumeration of three quadrant walks with small ste…
Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to…
In the field of enumeration of weighted walks confined to the quarter plane, it is known that the generating functions behave very differently depending on the chosen step set; in practice, the techniques used in the literature depend on…
In the past decade, a lot of attention has been devoted to the enumera-tion of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear…
In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the development of several original approaches, and led to a rich…
In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use…
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 are 2^8 problems of this type, but some…
In the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been intensely studied. As a result, the generating functions…
We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on $\mathbb{Z}^2$ defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or $-1$. We concern ourselves with…
The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this…
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, the kernel has genus one, and the step set…
We study planar walks that start from a given point (i\_0, j\_0), take their steps in a finite set S, and are confined in the first quadrant of the plane. Their enumeration can be attacked in a systematic way: the generating function Q(x,…
We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are…
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…
The enumeration of small steps walks confined to the first quadrant of the plane has attracted a lot of attention over the past fifteen years. The associated generating functions are trivariate formal power series in $x,y,t$ where the…
Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple hypergeometric form. In the following decade, this problem…
Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i belongs to S for all i, and none of the points w_i, i>0, lie on the half-line H= {(k,0):…
In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of…
Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in $\{0, \pm 1\}^2$: the generating function is D-finite if…
We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey \emph{two-step rules,} which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a…
Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of…