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In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.

This paper introduces Riordan-Krylov matrices. These matrices naturally generalize Riordan matrices by using Krylov matrices and a more general class of algebras in place of formal power series. Fundamental properties of Riordan-Krylov matrices that are analogous to those of Riordan matrices are developed. These results and techniques are used to study incidence algebras of a poset as well as the chain and zeta-multichain polynomials of the poset. Throughout the paper applications to enumeration problems in combinatorics and number theory are provided.

In this paper three new classes of matrices that have a computationally simple inverse are studied. Specifically, semi-involutory matrices are characterized for small orders and many interesting results are provided. Semi-involutory matrices can be thought of as a generalization of involutory matrices. In addition, signed semi-involutory matrices are studied and a natural generalization is introduced.

The symmetric Pascal matrix is a square matrix whose entries are given by binomial coefficients modulo 2. In 1997, Christopher and Kennedy defined and studied the binomial graph, which is the graph...

Given disjoint non-empty subsets S and T of {1,…,n−1}, a digraph D with the vertex set {1,2,…,n} is called a directed Toeplitz graph provided the arc i→j occurs if and only if i−j∈S or j−i∈T. We in...

Asymptotic behavior of the Mertens function M ( n ) = ∑ k = 0 n μ ( k ) which is equal to the determinant of the n × n Redheffer matrix, is known to be closely related to the Riemann hypothesis. An infinite matrix whose nth leading principal minor is equal to M ( n ) for all n ≥ 1 is called a Mertens equimodular matrix. We use Riordan matrices to find a large class of Mertens equimodular matrices, each element of which is called by us a Riordan-Redheffer matrix, briefly an R-R matrix. We also give the generating function for the characteristic polynomials of R-R matrices. As a result, we introduce several examples of R-R matrices that reveal interesting spectral properties. Further, we pose two conjectures on the eigenvalues of those R-R matrices. Finally, we find a sufficient condition for the Riemann hypothesis using the smallest singular value of a R-R matrix.

In this paper, we generalize the Laguerre polynomials in terms of q-analogue for Riordan matrices. To be more specific, for α ∈ N 0 , we introduce new q-Laguerre polynomials L n ( α ) ( x ; q ) by defining the Eulerian generating function for L n ( α ) ( x ; q ) as ( ∏ j = 0 α 1 1 + q j + 1 z ) e q [ x z 1 + z ] . Interestingly, it turns out that L n ( α ) ( x ; q ) have combinatorial descriptions in the aspect of the inversions of factorized permutations and q-rook numbers. We locate their zeros and develop their algebraic properties as well.

We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line $x=1/2+it$.

Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix $A$ is an $n\times n$ pseudo-involution then the singular values of $A$ must come in reciprocal pairs. Moreover, we give a complete analysis of existence and nonexistence of the eigenvectors of Riordan matrices. This leads to a surprising partition of the group of Riordan matrices into matrices with three different types of eigenvectors. Finally, given a nonzero vector $v$, we investigate the Riordan matrices $A$ that stabilize the vector $v$, t i.e. $Av=v$., Comment: 16 pages

Distinct letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word of the form xyxy... (of even or odd length) or a word of the form yxyx... (of even or odd length). A simple graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E. Thus, edges of G are defined by avoiding the consecutive pattern 11 in a word representing G, that is, by avoiding xx and yy. In 2017, Jeff Remmel introduced the notion of a k-11-representable graph for a non-negative integer k, which generalizes the notion of a word-representable graph. Under this representation, edges of G are defined by containing at most k occurrences of the consecutive pattern 11 in a word representing G. Thus, word-representable graphs are precisely 0-11-representable graphs. Our key result in this paper is showing that every graph is 2-11-representable by a concatenation of permutations, which is rather surprising taking into account that concatenation of permutations has limited power in the case of 0-11-representation. Also, we show that the class of word-representable graphs, studied intensively in the literature, is contained strictly in the class of 1-11-representable graphs. Another result that we prove is the fact that the class of interval graphs is precisely the class of 1-11-representable graphs that can be represented by uniform words containing two copies of each letter. This result can be compared with the known fact that the class of circle graphs is precisely the class of 0-11-representable graphs that can be represented by uniform words containing two copies of each letter.

In this paper, we show that an infinite lower triangular matrix A = [ a i j ] i , j ∈ N 0 is an exponential Riordan matrix A = E ( g , f ) given by ∑ i ≥ j a i j z i / i ! = g f j / j ! if and only if there exist both a horizontal pair { h n ; h ˜ n } n ≥ 0 and a vertical pair { v n ; v ˜ n } n ≥ 0 of sequences that represent all the elements in the matrix. As a consequence, we obtain that if the horizontal and vertical pairs of an exponential Riordan matrix are identical then the matrix is an involution. In addition, this concept can be applied to obtain the determinants of the production matrix and some conditions for the d -orthogonality of the Sheffer polynomial sequences.

We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural linear action induced on the Frechet space K N by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K ( ( x ) ) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper [18] where the Riordan group was described using inverse sequences of groups of finite matrices.

We obtain spectral properties of the Pascal graphs by exploring its spectral graph invariants such as the algebraic connectivity, the first three largest Laplacian eigenvalues and the nullity. Some open problems pertaining to the Pascal graphs are given.

We first consider two groups, F 0 = { g ∈ C [ [ z ] ] | g ( 0 ) ≠ 0 } under multiplication and F 1 = z F 0 under composition, where C [ [ z ] ] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F 0 ⋊ F 1 . It may be viewed as a group extension of F 1 by F 0 . In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F 0 . This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums.

Riordan bases in a power series ring of several variables are introduced as a special type of Schauder bases. As a part of the method of generating differentials, interplay of Riordan bases generalizes Lagrange inversion formulas. The set of Riordan bases has a group law with a specified set of variables as the unit element. Riordan arrays appear in a representation of the group.

We consider ordered trees with a distinguished vertex which we call a mutator. There are many situations where this model arises. An ordered tree could represent a river network, supply lines, an employee organization chart, a phylogenetic tree, or a family tree. The mutator could be a dam or a source of pollution, a break in a supply line, a corrupt employee, and so on. We look at such things as the number of affected vertices, distance from the root, and the effect of various succession rules. Among the types of trees these results apply to are ordered trees defined by a uniform updegree requirement. We find some profiles of trees with a mutation in terms of Riordan matrices. The asymptotic formulas for the ratios of vertices of the new type to all vertices are also derived by using singularity analysis of generating functions.