Mixed succession rules: the commutative case

TL;DR

This paper studies mixed succession rules in combinatorics, focusing on the case where two production rules commute, and provides formulas linking the resulting sequences to the component rules.

Contribution

It introduces a systematic approach to analyze mixed succession rules with commuting operators, deriving a general formula for their associated sequences.

Findings

Derived a formula relating mixed succession rule sequences to component sequences.

Demonstrated the approach with illustrative examples.

Enhanced understanding of combinatorial structures with multiple production rules.

Abstract

We begin a systematic study of the enumerative combinatorics of mixed succession rules, which are succession rules such that, in the associated generating tree, the nodes are allowed to produce their sons at several different levels according to different production rules. Here we deal with a specific case, namely that of two different production rules whose rule operators commute. In this situation, we are able to give a general formula expressing the sequence associated with the mixed succession rules in terms of the sequences associated with the component production rules. We end by providing some examples illustrating our approach.