Generating trees and proper Riordan Arrays

The Tennis Ball Problem

“…The correspondence between tennis ball s-sequences and generating trees allows us to use results obtained for these latter combinatorial objects. First of all, we re-introduce from [6] the concept of an AGT matrix. Definition 4.1.…”

“…This immediately gives d 0 ðtÞ ¼ TðtÞ: The generating function of the other columns is given by the following: Proof. First, we observe that specification (2.2) can be written in the product notation (see [6]): ðkÞ ! Q kþsÀ1 j¼1 ðk þ s À jÞ; s52; and this immediately shows that a node at level n þ 1 with label k þ 2 can be determined from the nodes at level n with label h þ 1;…”

Decomposition and Group Theoretic Characterization of Pairs of Inverse Relations of the Riordan Type

“…The group is quite easily developed but unifies many themes in enumeration, including of the generalized concept of Renewal Array defined by Rogers in 1978 ( [28] and [1,[19][20][21], etc.). Their basic idea was to define a class of infinite lower triangular arrays with properties analogous to those those of the Pascal triangle whose elements.…”

Proper generating trees and their internal path length

“…The infinite triangles of Pascal, Catalan, Motzkin and Schröder are important and meaningful examples of pRa's, and many others have been proposed and developed (see, e.g., [10,18,20]). In the recent literature, Riordan arrays have attracted the attention of various authors from many point of view and many examples and applications can be found (see, e.g., [1,6,11,[13][14][15]17,24,21] but the literature is still developing now). In particular, Merlini and Verri [15] pointed out an important connection between pRa's and generating trees and they call proper generating trees the corresponding trees.…”

“…In the recent literature, Riordan arrays have attracted the attention of various authors from many point of view and many examples and applications can be found (see, e.g., [1,6,11,[13][14][15]17,24,21] but the literature is still developing now). In particular, Merlini and Verri [15] pointed out an important connection between pRa's and generating trees and they call proper generating trees the corresponding trees. There exists a vast literature about the concept of generating trees: it was used for the first time, without any specific name, by Chung et al [5] and successively by West [22,23].…”