Gi-Sang Cheon

A note on the Bernoulli and Euler polynomials

In this paper, we obtain a simple property of the Bernoulli polynomials Bn(x) and the Euler polynomials En(x). As a consequence, the relationship between two polynomials is obtained from bn(ξ)=∑k=0 k≠1nnkBk(0)En−k(ξ).

Stirling matrix via Pascal matrix

Abstract The Pascal-type matrices obtained from the Stirling numbers of the first kind s(n,k) and of the second kind S(n,k) are studied, respectively. It is shown that these matrices can be factorized by the Pascal matrices. Also the LDU-factorization of a Vandermonde matrix of the form V n (x,x+1,…,x+n−1) for any real number x is obtained. Furthermore, some well-known combinatorial identities are obtained from the matrix representation of the Stirling numbers, and these matrices are generalized in one or two variables.

r-Whitney numbers of Dowling lattices

Abstract Let G be a finite group of order m ≥ 1 . A Dowling lattice Q n ( 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 Q n ( 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.

Combinatorics of Riordan arrays with identical A and Z sequences

Abstract In theory, Riordan arrays can have any A -sequence and any Z -sequence. For examples of combinatorial interest they tend to be related. Here we look at the case that they are identical or nearly so. We provide a combinatorial interpretation in terms of weighted Łukasiewicz paths and then look at several large classes of examples.

Protected points in ordered trees

In this note we start by computing the average number of protected points in all ordered trees with n edges. This can serve as a guide in various organizational schemes where it may be desirable to have a large or small number of protected points. We will also look a few subclasses with a view to increasing or decreasing the proportion of protected points.

Factorial Stirling matrix and related combinatorial sequences

Abstract In this paper, some relationships between the Stirling matrix, the Vandermonde matrix, the Benoulli numbers and the Eulerian numbers are studied from a matrix representation of k!S(n,k) which will be called the factorial Stirling matrix, where S(n,k) are the Stirling numbers of the second kind. As a consequence a number of interesting and useful identities are obtained.

A generalization of Lucas polynomial sequence

In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.

GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION

In this paper, we obtain important combinatorial identities of generalized harmonic numbers using symmetric polynomials. We also obtain the matrix representation for the generalized harmonic numbers whose inverse matrix can be computed recursively.

RANK AND PERIMETER PRESERVERS OF BOOLEAN RANK-1 MATRICES

For a Boolean rank-1 matrix A = ab t , we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the Boolean linear operators that preserve rank and perimeter of Boolean rank-1 matrices.

Matrices determined by a linear recurrence relation among entries

Abstract A matrix A =[ a ij ] is called a 7-matrix if its entries satisfy the recurrence relation αa i −1, j −1 + βa i −1, j = a ij where α , β are fixed numbers. A 7-matrix is completely determined by its first row and first column. In this paper we determine the structure of 7-matrices and investigate the sequences represented by columns of infinite 7-matrices.