Proper generating trees and their internal path length

Abstract

We find the generating function counting the total internal path length of any proper generating tree. This function is expressed in terms of the functions (d(t), h(t)) defining the associated proper Riordan array. This result is important in the theory of Riordan arrays and has several combinatorial interpretations.

Some New Binomial Sums Related to the Catalan Triangle

“…The Catalan triangles B and C often arise as examples of the infinite matrix associated to generating trees [7,12,34,36]. In the theory of Riordan arrays [45,46,49], much interest has been taken in the three triangles A, B and C, see [2,14,15,16,17,32,34,48,51]. In fact, A, B and C are Riordan arrays A = (C(t), tC(t) 2 ), B = (C(t) 2 , tC(t) 2 ), and C = (C(t), tC(t)), the electronic journal of combinatorics 21(1) (2014), #P1.…”

A symbolic treatment of Riordan arrays

“…Expanding r n (x), we may easily check that the coefficient of x k in [s r] n (x) is the (n, k)-entry of the Riordan array (γ, α)(σ, ρ). Hence, by (15), the polynomial sequence [s r] n (x) is the Sheffer sequence of (γ + σ.β.α D , α + ρ.β.α D ). Also, it readily follows that the Sheffer sequence x n of (ε, ε) is the identity element for the umbral composition and that s n (x) is the inverse of r n (x) if and only if (σ, ρ) = (L γ,α , L α ).…”

“…Riordan arrays form a group under matrix multiplication. The literature about Riordan arrays is vast and still growing and the applications cover a wide range of subjects, such as enumerative combinatorics, combinatorial sums, recurrence relations and computer science, among other topics [5,13,14,15,16,17,18,31,32,33,35]. Formally, ordinary Riordan arrays are a formulation of the 1-umbral calculus, whereas exponential Riordan arrays are a formulation of the n!-umbral calculus.…”

Four transformations on the Catalan triangle

“…The Catalan triangles B and C often arise as examples of the infinite matrix associated to a generating tree [4,22,24]. In the theory of Riordan arrays [31,32,35], much interest has been taken in the three triangles A, B and C, see [7,8,9,10,20,22,34,36]. In fact, A, B and C are Riordan arrays A = (C(t), tC(t) 2 ), B = (C(t) 2 , tC(t) 2 ), and C = (C(t), tC(t)),…”