Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Diagonals in Riordan matrices and applications

Gi-Sang Cheon, Ana Luzón, Manuel A. Morón, José L. Ramírez

Abstract

We introduce a method for describing Riordan matrices via recurrence relations along their diagonals. This provides a new structural description that complements the classical row-wise and column-wise constructions via the A-sequence. As an application, we characterize families of palindromic Riordan polynomials, yielding a new combinatorial interpretation in terms of lattice paths.

Diagonals in Riordan matrices and applications

Abstract

We introduce a method for describing Riordan matrices via recurrence relations along their diagonals. This provides a new structural description that complements the classical row-wise and column-wise constructions via the A-sequence. As an application, we characterize families of palindromic Riordan polynomials, yielding a new combinatorial interpretation in terms of lattice paths.
Paper Structure (15 sections, 124 equations, 2 figures)

This paper contains 15 sections, 124 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Motivation
  3. Some preliminaries results
  4. Some motivating examples
  5. Diagonals: a recurrence formula and examples
  6. A recurrence formula for diagonals of Riordan matrices
  7. Diagonals of the Pascal's triangle.
  8. Diagonals of Toeplitz matrices.
  9. Diagonals of the Catalan triangle
  10. Diagonals of the elements of the groups ${\mathcal{G}}_k$
  11. Diagonals of two particularly related triangles.
  12. Diagonals of $q$-cones.
  13. Palindromic Riordan polynomials and a combinatorial interpretation
  14. A combinatorial interpretation
  15. Acknowledgements

Figures (2)

  • Figure 1: Equivalent Delannoy paths.
  • Figure 2: The $13$ Delannoy paths in ${\mathcal{D}}(2,2)$, grouped into equivalence classes under $\sim$.

Theorems & Definitions (14)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 4 more