Catalan-like numbers and succession rules

TL;DR

This paper establishes a connection between the ECO method and Catalan-like numbers using linear algebra, enabling translation of succession rules into Aigner matrices and exploring their interactions.

Contribution

It introduces a linear algebra framework to relate succession rules with Catalan-like numbers via basis changes in polynomial spaces.

Findings

Succession rules can be transformed into Aigner matrices through basis changes.

The framework applies to specific classes of succession rules.

Examples illustrate the translation process and potential for further study.

Abstract

The ECO method and the theory of Catalan-like numbers introduced by Aigner seems two completely unrelated combinatorial settings. In this work we try to establish a bridge between them, aiming at starting a (hopefully) fruitful study on their interactions. We show that, in a linear algebra context (more precisely, using infinite matrices), a succession rule can be translated into a (generalized) Aigner matrix by means of a suitable change of basis in the vector space of one-variable polynomials. We provide some examples to illustrate this fact and apply it to the study of two particular classes of succession rules.