Related papers: North-East Lattice Paths Avoiding $k$ Collinear Po…
We enumerate the number of monotonic lattice paths starting at $(0,0)$ and terminating at $(m,n)$ in which $l$ of the first $k$ steps lie below the line $y=x\ (0\leq k\leq m\leq n)$. These closed formulas consist of terms which are a…
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…
Finding paths in graphs is a fundamental graph-theoretic task. In this work, we we are concerned with finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident…
The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek…
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…
We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \geq 0, y \geq 0$ of the square-lattice with unit steps in the…
We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph $G$ with two distinct terminal vertices and two positive integers $p$ and $k$, the question is…
Lattice paths are functional entities that model efficient navigation in discrete/grid maps. This paper presents a new scheme to generate collision-free lattice paths with utmost efficiency using the bijective property to rooted ordered…
Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least $k$ vertices is considered long. When $k \le 3$, the…
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 begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal…
A prototypical problem on which techniques for exact enumeration are tested and compared is the enumeration of self-avoiding walks. Here, we show an advance in the methodology of enumeration, making the process thousands or millions of…
Consider lattice paths in Z^2 taking unit steps north (N) and east (E). Fix positive integers r,s and put an equivalence relation on points of Z^2 by letting v,w be equivalent if v - w = m (r,s) for some m in Z. Call a lattice path valid if…
Let $P$ be a set of $n$ points in the plane, each point moving along a given trajectory. A {\em $k$-collinearity} is a pair $(L,t)$ of a line $L$ and a time $t$ such that $L$ contains at least $k$ points at time $t$, the points along $L$ do…
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.…
We study a model of semi-flexible self-avoiding trails, where the lattice paths are constrained to visit each lattice edge at most once, with configurations weighted by the number of collisions, crossings and bends, on a Husimi lattice…
The regularization path of the Lasso can be shown to be piecewise linear, making it possible to "follow" and explicitly compute the entire path. We analyze in this paper this popular strategy, and prove that its worst case complexity is…
In this paper we address the speed planning problem for a vehicle along a predefined path. A weighted sum of two conflicting objectives, energy consumption and travel time, is minimized. After deriving a non-convex mathematical model of the…
We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding…
The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces `short' simplex paths from any given vertex to an optimal one. We consider a lattice polytope $P$ contained in $[0,k]^n$ and…