Mixed succession rules: The commutative case

We begin a systematic study of the enumerative combinatorics of mixed succession rules, i.e. succession rules such that, in the associated generating tree, nodes are allowed to produce 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 rule in terms of the sequences associated with the component production rules. We end by providing examples illustrating our approach.