Riordan-Krylov matrices over an algebra Cheon, Gi-Sang; Curtis, Bryan; Shader, Bryan
Linear algebra and its applications,
03/2022, Volume:
636
Journal Article
Peer reviewed
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.
We present combinatorial and analytical results concerning a Sheffer sequence with an exponential generating function of the form G(s,z)=eczs+αz2+βz4, where α,β,c∈R with β<0 and c≠0. We demonstrate ...that the zeros of all polynomials in such a Sheffer sequence are either real, or purely imaginary. Additionally, using the properties of Riordan matrices we show that our Sheffer sequence satisfies a four-term recurrence relation of order 4, and we also exhibit a connection between the coefficients of these Sheffer polynomials and the number of nodes with a given label in certain marked generating trees.
A conjecture on minimum permanents Cheon, Gi-Sang; Song, Seok-Zun
Czechoslovak mathematical journal,
04/2024, Volume:
74, Issue:
1
Journal Article
Peer reviewed
We consider the permanent function on the faces of the polytope of certain doubly stochastic matrices, whose nonzero entries coincide with those of fully indecomposable square (0, 1)-matrices ...containing the identity submatrix. We show that a conjecture in K. Pula, S. Z. Song, I. M. Wanless (2011), is true for some cases by determining the minimum permanent on some faces of the polytope of doubly stochastic matrices.
We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form G(x,z)=Q(z)xQ(−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.
In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring B={0,1}. Given subsets S and T of {1,…,n−1}, an n×n Toeplitz matrix A=Tn〈S;T〉 is ...defined to have 1 as the (i,j)-entry if and only if j−i∈S or i−j∈T. We show that if maxS+minT≤n and minS+maxT≤n, then A has the matrix period d/d′ and the competition period 1 where d=gcd(s+t|s∈S,t∈T) and d′=gcd(d,minS). Moreover, it is shown that the limit of the matrix sequence {Am(AT)m}m=1∞ 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.
Symmetric Pascal matrices and related graphs Cheon, Gi-Sang; Kim, Jang Soo; Mojallal, Seyed Ahmad ...
Linear & multilinear algebra,
12/2022, Volume:
70, Issue:
21
Journal Article
Peer reviewed
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 ...whose adjacency matrix is the symmetric Pascal matrix. They computed the spectrum of the binomial graph of order a power of 2. In this paper, we study spectral properties of the binomial graph of any order such as eigenvalues and eigenvectors, algebraic connectivities and inertia indices. We also compute the determinant of the symmetric Pascal matrix in modulo 3.
Let G be a finite group of order m≥1. A Dowling lattice Qn(G) is the geometric lattice of rank n over G. In this paper, we define the r-Whitney numbers of the first and second kind over Qn(G), ...respectively. This concept is a common generalization of the Whitney numbers and the r-Stirling numbers of both kinds. We give their combinatorial interpretations over the Dowling lattice and we obtain various new algebraic identities. In addition, we develop the r-Whitney–Lah numbers and the r-Dowling polynomials associated with the Dowling lattice.
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×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 sets of eigenvectors. Finally, given a nonzero vector v, we investigate the Riordan matrices A that stabilize the vector v, i.e.Av=v.
In this paper, we give a new angle to interpret Riordan arrays by showing that every Riordan array can be expressed as a Krylov matrix. We then use this idea to obtain some groups containing the ...Riordan group as a subgroup. Moreover, we study Lie algebras for the extended Riordan groups as Lie groups.
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