We use the Lagrange inversion theorem to characterize the central coefficients of matrices in the Bell subgroup of the Riordan group of matrices. We give examples of how by using different means of calculating these coefficients we can deduce the generating functions of interesting sequences.

We use the formalism of the Riordan group to study a one-parameter family of lower-triangular matrices related to the weight distribution of maximum distance separable codes. We obtain factorization results for these matrices. We then derive alternative expressions for the weight distribution of MDS codes. We define related weight ratios and show that they satisfy a certain linear recurrence.

Zhang D1, Fernon A1, Lam B1 and Riordan SM1,2* Author Affiliations 1Gastrointestinal Liver Unit, Prince of Wales Hospital, Australia 2Prince Clinical School, UNSW Medicine Health, University New South Wales, Received: August 23, 2022 | Published: September 06, Corresponding author: Stephen Riordan, Senior Staff Specialist Head, Gastrointestinal Unit Hospital Barker Street, Randwick NSW 2031, DO...

As a variation of De Bruijn graphs on strings of symbols, the graph of overlapping permutations has a directed edge π(1)π(2) . . . π(n+1) from the standardization of π(1)π(2) . . . π(n) to the standardization of π(2)π(3) . . . π(n + 1). In this paper, we consider the enumeration of d-cycles in the subgraph of overlapping (231, 41̄32)avoiding permutations. To this end, we introduce the notions of...

We consider an identity relating Fibonacci numbers to Pascal’s triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a ...

We define a generalization of the Stirling numbers of the second kind, which depends on two parameters. The matrices of integers that result are exponential Riordan arrays. We explore links to orthogonal polynomials by studying the production matrices of these Riordan arrays. Generalized Bell numbers are also defined, again depending on two parameters, and we determine the Hankel transform of t...

We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobás-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobás-Riordan polynomials.

In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results, both in finding closed expressions for sums and, especially, in solving classes of combinatorial sum inversions.

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called p, q -Fibonacci polynomials. We obtain combinatorial identities and by using Riordanmethodwe get factorizations of Pascal matrix involvin...

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.