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Notes on Riordan arrays and lattice paths

Paul Barry

TL;DR

The paper investigates deep links between Riordan arrays and lattice-path enumeration, framing path counts with Riordan generating functions $A(x,y)=\frac{g(x)}{1-y f(x)}$. It develops rectification and triangulation of Riordan arrays, uses the A-matrix and various characterizations, and analyzes left-factors for diverse step sets including downward moves. The key contributions include concrete connections for Delannoy, Catalan, Dyck, Motzkin, and Schröder families, equinumerous path results, and a unified A-matrix framework for deriving recurrences and generating functions. This work provides a versatile toolkit for counting lattice paths with colored or generalized steps, with potential extensions to Fuss-Catalan families and colored-path variants.

Abstract

In this note, we explore links between Riordan arrays and lattice paths. We begin by describing Riordan arrays, and some of their generalizations, including rectifications and triangulations. We the consider Riordan array links to lattice paths with steps of type $(a,b)$, where $a$ and $b$ are nonnegative. We consider common Riordan arrays that are linked to lattice paths, as well as showing links between almost Riordan arrays and lattice paths. We then consider lattice paths with step sets that include downward steps, and show how the $A$-matrix characterization of Riordan arrays plays a key role in analysing corresponding Riordan arrays.

Notes on Riordan arrays and lattice paths

TL;DR

The paper investigates deep links between Riordan arrays and lattice-path enumeration, framing path counts with Riordan generating functions . It develops rectification and triangulation of Riordan arrays, uses the A-matrix and various characterizations, and analyzes left-factors for diverse step sets including downward moves. The key contributions include concrete connections for Delannoy, Catalan, Dyck, Motzkin, and Schröder families, equinumerous path results, and a unified A-matrix framework for deriving recurrences and generating functions. This work provides a versatile toolkit for counting lattice paths with colored or generalized steps, with potential extensions to Fuss-Catalan families and colored-path variants.

Abstract

In this note, we explore links between Riordan arrays and lattice paths. We begin by describing Riordan arrays, and some of their generalizations, including rectifications and triangulations. We the consider Riordan array links to lattice paths with steps of type , where and are nonnegative. We consider common Riordan arrays that are linked to lattice paths, as well as showing links between almost Riordan arrays and lattice paths. We then consider lattice paths with step sets that include downward steps, and show how the -matrix characterization of Riordan arrays plays a key role in analysing corresponding Riordan arrays.

Paper Structure

This paper contains 11 sections, 4 theorems, 194 equations, 3 figures.

Table of Contents

  1. Preliminaries on Riordan arrays
  2. Characterizations of Riordan arrays
  3. Lattice paths and Riordan arrays
  4. Triangulating rectified Riordan arrays
  5. Dyck paths, Motzkin paths and Catalan matrices
  6. Almost Riordan arrays
  7. Downward steps
  8. A result on equinumerous lattice paths
  9. Using the $A$-matrix
  10. Two matrices
  11. Conclusions

Key Result

Theorem 5

A lower-triangular array $(t_{n,k})_{0 \le n,k \le \infty}$ is a Riordan array if and only if there exists another array $A=(a_{i,j})_{i,j \in \mathbb{N}_0}$ with $a_{0,0} \ne 0$, and a sequence $(\rho_j)_{j \in \mathbb{N}_0}$ such that

Figures (3)

  • Figure 1: The Delannoy triangle $\left(\frac{1}{1-x},\frac{x(1+x)}{1-x}\right)$ as a path matrix
  • Figure 2: The Catalan matrix $(c(x),xc(x))$ as a path matrix
  • Figure 3: The ternary matrix $(t(x), xt(x))$ as a path matrix

Theorems & Definitions (29)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 5
  • Proposition 6
  • proof
  • Example 7
  • Example 8
  • Example 9
  • ...and 19 more