Level Generating Trees and Proper Riordan Arrays

The concept of generating trees has been introduced in the literature by Chung, Graham, Hoggat and Kleiman in [4] to examine Baxter permutations. This technique has been successfully applied by West [17, 18] to other classes of permutations and more recently to some other combinatorial classes such as plane trees and lattice paths (see Barcucci et al. [2]). In all these cases, a generating tree is associated to a certain combinatorial class, according to some enumerative parameter, in such a way that the number of nodes appearing on level n of the tree gives the number of n-sized objects in the class. If a problem has been defined by means of a generating tree, some device has to be used to obtain counting information on the objects of the associated combinatorial class. In [11] and [9], Merlini, Sprugnoli and Verri have introduced the concept of matrix associated to a generating tree (AGT matrix, for short): this is an infinite matrix (dn,k)n,k∈N where dn,k is the number of nodes at level n with label k+c, c being the root label. The main result in [9] is Theorem 3.3 (3.1 in this paper) which states the conditions under which an AGT matrix is a proper Riordan array, and vice versa. For example, the following labeled tree (this example concerns only non-marked nodes)