Riordan Arrays Research Articles

Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in [Formula: see text] starting at the origin made of steps [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], where two down-steps cannot be consecutive. We enumerate MAP and their prefixes avoiding peaks (respectively, valleys, respectively, double rise) according to the length, the type of the last step, and the height of its end-point. We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzkin paths.

Enumeration of the Motzkin paths above a line of rational slope

We introduce Lm-Motzkin paths similar to m-Dyck paths. For fixed positive integer m, Lm-Motzkin paths are a generalization of Motzkin paths that start at (0,0), use steps U=(1,1), D=(1,−1) and L=(1,0), remain weakly above the line y=m−1mx, and end on this line. The number Mn(m) of Lm-Motzkin paths running from (0,0) to (mn,(m−1)n) is called the n-th Lm-Motzkin number. We first prove that the generating function Mm(t) of the Lm-Motzkin numbers satisfies the equation Mm(t)=1+tMm(t)m+t2Mm(t)2m. We then use this generating function and Riordan array to discuss the enumerations of the partial Lm-Motzkin paths and the partial grand Lm-Motzkin paths. Finally, we consider the colored Lm-Motzkin paths, and present two classes of γ-positive polynomials via enumerating these paths.

Leonardo and hyper-Leonardo numbers via Riordan arrays

UDC 511 A generalization of the Leonardo numbers is defined and called the hyper-Leonardo numbers. Infinite lower triangular matrices, whose elements are Leonardo and hyper-Leonardo numbers are considered. Then the A - and Z -sequences of these matrices are obtained. Finally, the combinatorial identities between the hyper-Leonardo and Fibonacci numbers are obtained using the fundamental theorem of the Riordan arrays.

Riordan Arrays and Difference Equations of Subdiagonal Lattice Paths

Riordan Arrays and Difference Equations of Subdiagonal Lattice Paths

A Riordan group poset

In this paper, we look at Riordan arrays with nonnegative integer entries. In the set, if AT=B for a nonnegative integer matrix T, then we say A≤B. Our primary focus centers around six kinds of key arrays: Appell, Bell, two-Bell, Natural, Hankel, and Pseudo-involution. We examine relations among Hankel, Bell and two-Bell in general, while the other entries are treated case by case. We start by looking at individual cases related to the little and large Schröder numbers. Then we look at the families related to k-Catalan arrays, k-Schröder arrays, and k-Motzkin arrays. We often can give combinatorial interpretations. Along the way, several intriguing counterexamples occur.

Endomorphisms of the riordan group, some new facts

We consider endomorphisms of the Riordan group. This issue was recently studied by Petrullo and Senato who gave the characterization of these endomorphisms in terms of endomorphisms of the Appell and the Associated subgroup. Here, we show that some of the latter ones turn out to be identities. Additionally, we establish some facts concerning the action of these endomorphism on the diagonal Riordan arrays.

Some properties of combinatorial triangles related to Horadam polynomials

The Horadam polynomials unify many well-known polynomials, such as the Fibonacci polynomials, the Lucas polynomials, the Pell polynomials, the Jacobsthal polynomials and the Chebyshev polynomials. We investigate some properties of various combinatorial triangles related to Horadam polynomials, including their properties as almost-Riordan arrays and Riordan arrays, their total positivity, the real-rootedness of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems).

Consecutive pattern-avoidance in Catalan words according to the last symbol

We study the distribution of the last symbol statistics on the sets of Catalan words avoiding a consecutive pattern of length at most three. For each pattern p, we provide a bivariate generating function, where the coefficient cp(n,k) of xnyk in its series expansion is the number of length n Catalan words avoiding p and ending with the symbol k. We deduce recurrence relations or closed forms for cp(n, k) and we provide asymptotic approximations for the expectation of the last symbol on all Catalan words avoiding p. Finally, we characterize the sequence cp(n, k) using Riordan arrays.

Some γ-positive polynomials arising from enumerations of the pseudo Schröder paths

We introduce a notion of pseudo Schröder paths, which are counted by the Schröder numbers. Using the symbolic method and the Lagrange inversion formula, we provide an alternative proof that the descent polynomials of separable permutations are γ-positive. We also present several classes of γ-positive polynomials via enumerating the pseudo Schröder paths according to several combinatorial statistics by applying Riordan arrays.

On the halves of double and 3-dimensional Riordan arrays

The vertical half, horizontal half and skew half of a Riordan array D=(dn,k)n,k∈N have been deeply studied by Yang, He, Barry and others. In this paper, we concentrate on using compression to study various halves of generalized Riordan arrays, such as double Riordan arrays, j-th order Riordan arrays. As applications, in the various halves of 3-dimensional Riordan array cases, we derive some identities related to the classical sequences such as Fibonacci, Pell, Pell-Lucas, Jacobsthal-Lucas and Jacobsthal numbers.

The halves of a 3-dimensional Riordan array

For a Rirodan array, its vertical half and horizontal half are studied separately before. In this paper, we propose the (m1,s1,r,m2,s2,l)-halves of a 3-dimensional Riordan array, which are natural generalizations of the skew halves of a usual Riordan array in [21]. Let G=[gi,j,k]i,j,k≥0 be a 3-dimensional Riordan array, its (m1,s1,r,m2,s2,l)-halves H=[hi,j,k]i,j,k≥0 are defined byhi,j,k=g(m1+1)i+(s1−2)j+r,m1i+(s1−1)j+r,m2i+(s2−1)j+lk. We obtain that the (m1,s1,r,m2,s2,l)-halves of a 3-dimensional Riordan array are all 3-dimensional Riordan arrays. We also consider the (m1,s1,r,m2,s2,l)-antecedents of a 3-dimensional Riordan array.

Left multiplication operators on the Riordan group

In this paper, we introduce the left multiplication operators on the Riordan group, and carry out a systematic study on them. By using these operators, we obtain a family H of infinite Riordan subgroups. This family includes not only some well-known subgroups, but also the subgroup families introduced recently in literature. Fundamental properties of the left multiplication operators and the subgroup family H are presented. The Riordan involutions and pseudo-involutions, the stabilizer subgroups in family H, and the n-th commutator subgroup of any subgroup in family H are studied. Some further applications of the left multiplication operators are also provided. In particular, we study a class of left multiplication operators which are closely related to the diagonal translation operator and the [m]-complementary operator, and enrich the theory of complementary Riordan arrays. Moreover, we introduce and study the higher-level lifts of Riordan arrays as well as the vertical and horizontal 1/m Riordan arrays. As a result, it can be found that the left multiplication operators provide a unified approach to numerous subgroups, operators and transformations of the Riordan group.

Summations on the Diagonals of a Riordan Array and Some Applications

Summations on the Diagonals of a Riordan Array and Some Applications

Exponential-type extended Riordan arrays and reciprocity law for generalized Stirling numbers

We propose an exponential-type extended Riordan array (or exponential recursive matrix) with permitted negative indices using the Roman factorial. In this context, we establish a fundamental theory of the generalized Stirling numbers introduced by Hsu and Shiue [11]. Subsequently, we provide a lucid explanation for the reciprocity law among these numbers found by Choi et al. [5] and Maltenfort [19]. Moreover, our method reveals that interesting numbers emerge in the fourth quadrant of the proposed exponential recursive matrix.

D-Tribonacci Polynomials and Their Matrix Representations

In this study, we define d-Tribonacci polynomials. Some combinatorial properties of the d- Tribonacci polynomials with matrix representations are obtained with the help of Riordan arrays. In addition, d- Tribonacci number sequence, which is a new generalization of this number sequence, has been obtained by considering Pascal matrix. With the help of Pascal matrix, two kinds of factors of d-Tribonacci polynomials were found. Also, infinite d-Tribonacci polynomials matrix and the inverses of these polynomials were found.

Linear Trees, Lattice Walks, and RNA Arrays

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.

The skew halves of a Riordan array

Given a Riordan array (gn,k)n,k∈N, its vertical half (g2n−k,n)n,k∈N and horizontal half (g2n,n+k)n,k∈N are studied separately before. In the present paper, we introduce the skew (r,s)-halves of a Riordan array which are infinite lower triangular matrices with generic (n,k)-th entries g2n+(s−2)k+r,n+(s−1)k+r for n≥k≥0. This allows us to discuss the vertical half and horizontal half in a uniform context. We show that the skew halves of a Riordan array are all Riordan arrays. As applications, we find several new identities involving the Pascal matrix, and Catalan triangles by applying the skew halves. We also consider the inversion problems: given a Riordan array G, we can construct its Riordan antecedent H such that the (r,s)-half of H is equal to G.

Riordan Groups in Higher Dimensions

Abstract: The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix multiplication to two group actions on such series, namely formal (convolution) multiplication and formal composition. We define the analogous Riordan groups involving formal power series in several variables, and establish the analogue of the Fundamental Theorem in that context. We discuss related groups of Laurent series and pose some questions.

Some binomial identities related to the Catalan triangles and the halves of the Pascal matrix

In this paper, five binomial sums with two additional parameters are shown to be equipollent by using the Riordan array method and the generalized Catalan matrices, as well as the halves of the Pascal matrix.

Certain results on hybrid relatives of the Sheffer polynomials

The multi-variable special matrix polynomials have been identified significantly both in mathematical and applied frameworks. Due to its usefulness and various applications, a variety of its extensions and generalizations have been investigated and presented. The purpose of the paper is intended to study and emerge with a new generalization of Hermite matrix based Sheffer polynomials by involving integral transforms and some known operational rules. Their properties and quasi-monomial nature are also established. Further, these sequences are expressed in determinant forms by utilizing the relationship between the Sheffer sequences and Riordan arrays. An analogous study of these results is also carried out for certain members belonging to generalized Hermite matrix based Sheffer polynomials.