Abstract

The main purpose of this paper is to derive generating functions for the numbers of lattice paths running from (0, 0) to any (n, k) in \({\mathbb{Z} \times \mathbb{N}}\) consisting of four types of steps: horizontal H = (1, 0), vertical V = (0, 1), diagonal D = (1, 1), and sloping L = (–1, 1). These paths generalize the well-known Delannoy paths which consist of steps H, V, and D. Several restrictions are considered. However, we mainly treat with those which will be needed to get the generating function for the numbers R(n, k) of these lattice paths whose points lie in the integer rectangle \({\{(x, y) \in \mathbb{N}^2 : 0 \leq x \leq n, 0 \leq y \leq k\}}\). Recurrence relation, generating functions and explicit formulas are given. We show that most of considered numbers define Riordan arrays.

Highlights

  • We follow the notation of Lehner [9] and Chen et al [2]

  • A lattice path is a sequence of points in the integer lattice Z2

  • Let us consider the family of lattice paths running from (0, 0) to (n, k) and consisting of horizontal steps H = (1, 0) and vertical steps V = (0, 1)

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Summary

Introduction

We follow the notation of Lehner [9] and Chen et al [2]. A lattice path is a sequence of points in the integer lattice Z2. Let us consider the family of lattice paths running from (0, 0) to (n, k) and consisting of horizontal steps H = (1, 0) and vertical steps V = (0, 1). D(n, k) is the number of lattice paths running from (0, 0) to (n, k) consisting of horizontal H , vertical V and diagonal steps D = (1, 1). Let L denote the set of integer lattice points {(i, j) ∈ Z×N}. Π j ) of adjacent steps πi of four types: horizontal (1, 0), vertical (0, 1), diagonal (1, 1), and sloping (−1, 1), and whose set of points is a subset of I. Let (x, y) be a lattice point and π a lattice path, we write (x, y) ∈ π if (x, y) is a point of π

List of Variants
List of Generating Functions
The Base Case
The Right Triangle
Generalized Right Triangle Paths
The Isosceles Triangle
Central Numbers of the Base Case
General Cases
The Lattice L
The Lattice Ln
The First Approach
The Second Approach
Counting Paths by Length
Riordan Arrays
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