Related papers: Automatic Classification of Restricted Lattice Wal…
Trying to enumerate all of the walks in a 2D lattice is a fun combinatorial problem and there are numerous applications, from polymers to sports. Computers provide a wonderful tool for analyzing these walks; we provide a Maple package for…
We give a summary of recent progress on the signed area enumeration of closed walks on planar lattices. Several connections are made with quantum mechanics and statistical mechanics. Explicit combinatorial formulae are proposed which rely…
We present general algorithms (fully implemented in Maple) for calculations of various quantities related to constrained directed walks for a general set of steps on the square lattice in two dimensions. As a special case, we rederive…
We define a random walk problem which admits analytic results, on a class of infinite periodic lattices which are directed and colored. Our approach is motivated from the fact that such lattices arise in string theoretic constructs of…
Feller's book An Introduction to Probability Theory and Its Application discusses statistics corresponding to sequences of coin tosses, with a dollar being won or lost depending on the outcome of each toss. This is equivalent to analyzing…
Enumeration of planar lattice walks is a classical topic in combinatorics, at the cross-roads of several domains (e.g., probability, statistical physics, computer science). The aim of this paper is to propose a new approach to obtain some…
Counting integer solutions of linear constraints has found interesting applications in various fields. It is equivalent to the problem of counting lattice points inside a polytope. However, state-of-the-art algorithms for this problem…
We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary "core" process that has a regenerative…
This work considers lattice walks restricted to the quarter plane, with steps taken from a set of cardinality three. We present a complete classification of the generating functions of these walks with respect to the classes algebraic,…
We provide some first experimental data about generating functions of restricted lattice walks with small steps in NN^4.
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…
This work develops a methodical approach to counting of walks on cartesian products, biproducts, symmetric and exterior powers and bipowers, Schur operations, coverings and semicoverings of weighted graphs. For weight and root lattices of…
The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their…
We describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of $N$ steps is of order $3^{N/4}$ times a polynomial in $N$, and so the approach is…
We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating…
We provide a new strategy to compute the exponential growth constant of enumeration sequences counting walks in lattice path models restricted to the quarter plane. The bounds arise by comparison with half-planes models. In many cases the…
Recent work of the author connected several parking function enumeration problems to enumerations of Catalan paths with respect to certain weight functions that are expressed in terms of the ascent lengths. Motivated by this, we generalise…
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…
This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. The main difference to classical walks is that its nondeterministic steps consist of sets of steps from a predefined set such that all possible…
In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard…