Mixed succession rules: The commutative case

Abstract

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.

Binary words avoiding a pattern and marked succession rule

“…We remark that, from the above definition, a node labelled (k) has precisely k sons. In [1], a succession rule having this property is said to be consistent. However, we can also consider succession rules, introduced in [7], in which the value of a label does not necessarily represent the number of its sons, and this will be frequently done in the sequel.…”

“…Matrix (α i,j ) i,j∈N is called the A-matrix of the Riordan array. If P [0] (t), P [1] (t), P [2] (t), . .…”

“…In the following cases we consider a path η having the same property. The path ω ′ which kills ω is obtained by performing on µxγx j+1−h the following: add the path x h−1 by applying h − 1 times the mapping associated to (k) 1 (k + 1) of the first production of (15), add the path xx h by applying the mapping associated to (k) 1 (0 2 ) of the first production of (15), add the path x j−h ηx by applying consecutive and appropriate mappings of the first production of ( 15) and these mappings must be completed by performing the actions giving the label (0 1 ) in case of no marked points. The path ν in ω ′ is obtained as in ω.…”