Related papers: Bounce statistics for rational lattice paths

We introduce a new type of lattice path, called brick-wall lattice path, and we derive a formula which counts the number of paths on these lattices imposing certain restrictions on the Cartesian plane. Connections to the Fibonacci sequence,…

For positive integers $\alpha$ and $\beta$, we define an $(\alpha,\beta)$-walk to be any sequence of positive integers satisfying $w_{k+2}=\alpha w_{k+1}+\beta w_k$. We say that an $(\alpha,\beta)$-walk is $n$-slow if $w_s=n$ with $s$ as…

We count a large class of lattice paths by using factorizations of free monoids. Besides the classical lattice paths counting problems related to Catalan numbers, we give a new approach to the problem of counting walks on the slit plane…

We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the…

In this note we introduce a determinant and then give its evaluating formula. The determinant turns out to be a generalization of the well-known ballot and Fuss-Catalan numbers, which is believed to be new. The evaluating formula is proved…

A lattice path in $\mathbb{Z}^d$ is a sequence $\nu_1,\nu_2,\ldots,\nu_k\in\mathbb{Z}^d$ such that the steps $\nu_i-\nu_{i-1}$ lie in a subset $\mathbf{S}$ of $\mathbb{Z}^d$ for all $i=2,\ldots,k$. Let $T_{m,n}$ be the $m\times n$ table in…

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result extends well known enumerative formulae concerning lattice paths, and its derivation involves a classical…

\L{}ukasiewicz paths are lattice paths in $\Bbb{N}^2$ starting at the origin, ending on the $x$-axis, and consisting of steps in the set $\{(1,k), k\geq -1\}$. We give generating function and exact value for the number of $n$-length…

The combinatorics of certain osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. More specifically, the paths being considered have fixed start and end points on respectively the lower and right…

We derive a series of results on random walks on a d-dimensional hypercubic lattice (lattice paths). We introduce the notions of terse and simple paths corresponding to the path having no backtracking parts (spikes). These paths label…

Gessel's walks are the planar walks that move within the positive quadrant $\mathbb{Z}_{+}^{2}$ by unit steps in any of the following directions: West, North-East, East and South-West. In this paper, we find an explicit expression for the…

It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fu{\ss}-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps.…

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never…

In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not…

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the…

On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the rectangle. For each integer $k$ we ask for…

There is a strikingly simple classical formula for the number of lattice paths avoiding the line x = ky when k is a positive integer. We show that the natural generalization of this simple formula continues to hold when the line x = ky is…

Let $a,b$ be fixed positive coprime integers. For a positive integer $g$, write $W_k(g)$ for the set of lattice paths from the startpoint $(0,0)$ to the endpoint $(ga,gb)$ with steps restricted to $\{(1,0), (0,1)\}$, having exactly $k$…

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…

In two recent works \cite{BMM,BK}, it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quadrant and associated with a finite group of birational transformations are holonomic, and…