A Note on Riordan Arrays with Catalan Halves

TL;DR

This paper explores Riordan arrays whose vertical and horizontal halves are connected to Catalan matrices, revealing structural relationships and properties within combinatorial matrix theory.

Contribution

It introduces a novel analysis of Riordan arrays with halves related to Catalan matrices, expanding understanding of their structural properties.

Findings

Identifies specific relationships between Riordan array halves and Catalan matrices.

Provides new characterizations of Riordan arrays with Catalan-related halves.

Enhances the theoretical framework connecting Riordan arrays and Catalan combinatorics.

Abstract

For a lower triangular matrix $(t_{n,k})$ we call the matrices with respective entries $(t_{2n-k,n})$ and $(t_{2n,n+k})$ the vertical and the horizontal halves. In this note, we discuss Riordan arrays whose halves are closely related to the Catalan matrices.