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Generating functions of lattice paths

Rui Duarte, António Guedes de Oliveira

Abstract

We recall the main types of lattice paths, which are sequences in the lattice of integer coordinates points in the plane. We start with the fundamental central lattice paths and Dyck paths and proceed in elementary terms through recently introduced lattice paths. For every type, we consider the respective generating function. In fact, through our approach (via Riordan arrays), various entries of the On-Line Encyclopedia of Integer Sequences are unified, clarified, and simplified.

Generating functions of lattice paths

Abstract

We recall the main types of lattice paths, which are sequences in the lattice of integer coordinates points in the plane. We start with the fundamental central lattice paths and Dyck paths and proceed in elementary terms through recently introduced lattice paths. For every type, we consider the respective generating function. In fact, through our approach (via Riordan arrays), various entries of the On-Line Encyclopedia of Integer Sequences are unified, clarified, and simplified.
Paper Structure (5 sections, 1 theorem, 26 equations, 3 figures)

This paper contains 5 sections, 1 theorem, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Other lattice paths
  3. Central Delannoy paths and Schröder paths
  4. [Big] Motzkin paths
  5. Restricted central Delannoy and big Motzkin paths

Key Result

Lemma 1

Figures (3)

  • Figure 1: A central lattice path and a Dyck path
  • Figure 2: Delannoy and Schröder paths with $n=2$
  • Figure 3: Motzkin and big Motzkin paths

Theorems & Definitions (2)

  • Lemma 1
  • proof