Search Results (6)

Let $q_1+\cdots +q_n+m$ objects be arranged in n rows with $q_1,\dots ,q_n$ objects and one last row with m objects. The Janjić–Petković counting function denotes the number of $(n+k)$-insets, defined as subsets containing $n+k$ objects such that at least one object is chosen from each of the first n rows, generalizing the binomial coefficient that is recovered for $q_1$ = … = $q_n$ = 1, as then only the last row matters. Here, we discuss two explicit forms, combinatorial interpretations, recursion relations, an integral representation, generating functions, convolutions, special cases, and inverse pairs of summation formulas. Based on one of the generating functions, we show that the Janjić–Petković counting function, like the binomial coefficients that it generalizes, may be regarded as a Riordan array, leading to additional identities. As an application to a physical system, we calculate the heat capacity of a many-body system for which the configurations are constrained as described by the Janjić–Petković counting function, resulting in a modified Schottky anomaly. Full article

This paper investigates the use of Riordan arrays in the enumeration and transformation of lattice paths through a combinatorial framework of promotion. We demonstrate how Dyck paths can be promoted to generalised Motzkin and Schröder paths via two key transformations: the Binomial and Chebyshev transforms, each associated with specific Riordan arrays. These promotions yield classical integer sequences and continued fraction representations that enumerate weighted lattice paths. The framework is further extended to analyse grand paths, which are permitted to cross below the x-axis. We develop constructive bijections establishing explicit correspondences between promoted path families. The promotion framework offers new insights into known integer sequences and enables a unified approach to the generalisation and classification of lattice paths. Full article

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related to Riordan arrays, Sheffer matrices, and their q analogs. Full article

The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions of the idea of a Riordan array have included almost Riordan arrays, double Riordan arrays, and their generalizations. After giving a brief overview of the Riordan group, we define two further extensions of the notion of Riordan arrays, and we give a number of applications for these extensions. The relevance of these applications indicates that these new extensions are worthy of study. The first extension is that of the reverse symmetrization of a Riordan array, for which we give two applications. The first application of this symmetrization is to the study of a family of Riordan arrays whose symmetrizations lead to the famous Robbins numbers as well as to numbers associated with the 20 vertex model of mathematical physics. We provide closed-form expressions for the elements of these arrays, and we also give a canonical Catalan factorization for them. We also describe an alternative family of Riordan arrays whose symmetrizations lead to the same integer sequences. The second application of this symmetrization process is to the area of the enumeration of lattice paths. We remain with the applications to lattice paths for the second extension of Riordan arrays that we introduce, which is the interleaved Riordan array. The methods used include generating functions, linear algebra, weighted compositions, and linear recurrences. In the case of the symmetrization process applied to Riordan arrays, we focus on the principal minor sequences of the resulting square matrices in the context of integrable lattice models. Full article

Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout the literature, largely influenced by the author’s background and the specific application under consideration. Beyond the representation through recurrence relations, other approaches include those based on generating functions, explicit formulas, matrix exponentiation, the method of undetermined coefficients and several others. Among these, the generating function approach is particularly prevalent in enumerative combinatorics due to its versatility and widespread use. The primary objective of this work is to introduce an alternative representation grounded in the theory of Riordan arrays. This representation provides a general formula expressed in terms of the vectors of constants and initial conditions associated with any recurrence relation of a given order, offering a new perspective on the structure of such sequences. Full article

The leftmost column entries of RNA arrays I and II count the RNA numbers that are related to RNA secondary structures from molecular biology. RNA secondary structures sometimes have mutations and wobble pairs. Mutations are random changes that occur in a structure, and wobble pairs are known as non-Watson–Crick base pairs. We used topics from RNA combinatorics and Riordan array theory to establish connections among combinatorial objects related to linear trees, lattice walks, and RNA arrays. In this paper, we establish interesting new explicit bijections (one-to-one correspondences) involving certain subclasses of linear trees, lattice walks, and RNA secondary structures. We provide an interesting generalized lattice walk interpretation of RNA array I. In addition, we provide a combinatorial interpretation of RNA array II as RNA secondary structures with $n$ bases and $k$ base-point mutations where ω of the structures contain wobble base pairs. We also establish an explicit bijection between RNA structures with mutations and wobble bases and a certain subclass of lattice walks. Full article