The cyclic groups via the Pascal matrices and the generalized Pascal matrices

Let s be arbitrary integer, we introduce the notion of the matrix of type s, whose nonzero entries are the classical Horadam numbers . In this paper we consider singular case , then the Moore–Penrose inverse of the matrix is given. In the case , we obtain the Pseudoinverse of the generalized Fibonacci matrix . In addition, correlations between the matrix and the generalized Pascal matrices are discussed, and some combinatorial identities involving the Horadam numbers are derived.

The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices

Let F q be a finite field with q elements, n(≥2) a positive integer, and T n (q) the semigroup of all n × n upper triangular matrices over F q . The generalized Cayley graph GCay(T n (q)) of T n (q) is a directed graph with vertex set T n (q), in which there is a directed edge from a vertex A to a distinct vertex B if and only if B = XAY for some X, Y ∈ T n (q). The main result of this article proves that a bijective map σ is an automorphism of GCay(T n (q)) if and only if, for any vertex A of GCay(T n (q)), either σ(A) = P A AQ A or σ(A) = P A JA t JQ A , where A t denotes the transpose of A, , and P A and Q A are invertible upper triangular matrices depending on A.

A new algorithm for linear systems of the Pascal type

A word is a finite sequence of symbols. Parikh matrix of a word is an upper triangular matrix with ones in the main diagonal and nonnegative integers above the main diagonal which are counts of certain scattered subwords in the word. On the other hand, a picture array, which is a rectangular arrangement of symbols, is an extension of the notion of a word to two dimensions. Parikh matrices associated with a picture array have been introduced, and their properties have been studied. Here, we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness, and a restricted shuffle operator extending the corresponding notions studied in the case of words. We also obtain properties of Parikh matrices of arrays formed by certain geometric operations.

Since the undecidability of the mortality problem for 3 × 3 matrices over integers was proved using the Post Correspondence Problem, various studies on decision problems of matrix semigroups have emerged. The freeness problem in particular has received much attention but decidability remains open even for 2 × 2 upper triangular matrices over nonnegative integers. Parikh matrices are upper triangular matrices introduced as a generalization of Parikh vectors and have become useful tools in studying of subword occurrences. In this work, we focus on semigroups of Parikh matrices and study the freeness problem in this context.

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

In this paper, we study a balancing problem of ordinary binomial coefficients. The diop-hantine equation [Formula: see text] in positive integers x and y ≥ x + 2 will be investigated, where k and ℓ are given positive integers. In case of k < ℓ the basic tool is the Runge method. If k ≥ ℓ, then the main results state effective and non-effective finiteness theorems on the subject matter. Further, computational results are also reported including the complete solutions when 1 ≤ k, ℓ ≤ 3, and a computer search otherwise.

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

Catalan matrix and related combinatorial identities

Let us examine some properties of the q-binomial coefficients, defined by (4.5), with n and j being nonnegative integers and n ≥ j. Because we will recover the ordinary binomial coefficients if we take q → 1, we expect their q-analogues to have similar properties. Firstly, as already remarked in (5.4), $$ \left[ {\begin{array}{*{20}c} n j \end{array} } \right] = \frac{{[n]!}} {{[j]![n - j]!}} = \left[ {\begin{array}{*{20}c} n {n - j} \end{array} } \right] $$ (6.1) follows exactly the classical result. However, the correspondence is more subtle for another identity of binomial coefficients, the Pascal rule: $$ \left( {\begin{array}{*{20}c} n j \end{array} } \right) = \left( {\begin{array}{*{20}c} {n - 1} {j - 1} \end{array} } \right) + \left( {\begin{array}{*{20}c} {n - 1} j \end{array} } \right), 1 \leqslant j \leqslant n - 1. $$

The Raney numbers are a class of combinatorial numbers generalising the Fuss–Catalan numbers. They are indexed by a pair of positive real numbers (p, r) with p > 1 and 0 < r ⩽ p, and form the moments of a probability density function. For certain (p, r) the latter has the interpretation as the density of squared singular values for certain random matrix ensembles, and in this context equilibrium problems characterising the Raney densities for (p, r) = (θ + 1, 1) and (θ/2 + 1, 1/2) have recently been proposed. Using two different techniques—one based on the Wiener–Hopf method for the solution of integral equations and the other on an analysis of the algebraic equation satisfied by the Green's function—we establish the validity of the equilibrium problems for general θ > 0 and similarly use both methods to identify the equilibrium problem for (p, r) = (θ/q + 1, 1/q), θ > 0 and . The Wiener–Hopf method is used to extend the latter to parameters (p, r) = (θ/q + 1, m + 1/q) for m a non-negative integer, and also to identify the equilibrium problem for a family of densities with moments given by certain binomial coefficients.

Let $p>3$ be a prime and let $a$ be a positive integer. We show that if $p\equiv1\pmod 4$ or $a>1$ then $$\sum_{k=0}^{\lfloor\frac34p^a\rfloor}\frac{\binom{2k}k^2}{16^k}\equiv\l(\frac{-1}{p^a}\r)\pmod{p^3}$$ with $(-)$ the Jacobi symbol, which confirms a conjecture of Z.-W. Sun. We also establish the following new congruences: \begin{align*}\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{3k}k}{27^k}\equiv&\l(\frac p3\r)\frac{2^p+1}3\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{6k}{3k}\binom{3k}k}{(2k+1)432^k}\equiv&\l(\frac p3\r)\frac{3^p+1}4\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{4k}{2k}\binom{2k}k}{(2k+1)64^k}\equiv&\l(\frac{-1}p\r)2^{p-1}\pmod{p^2}. \end{align*} Note that in 2003 Rodriguez-Villeguez posed conjectures on $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2}{16^k},\ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k}{27^k},\ \sum_{k=1}^{p-1}\frac{\binom{4k}{2k}\binom{2k}k}{64^k},\ \sum_{k=1}^{p-1}\frac{\binom{6k}{3k}\binom{3k}k}{432^k}$$ modulo $p^2$ which were later proved.

For a finite system $$A = {\left\{ {a_{s} + n_{s} \mathbb{Z}} \right\}}^{k}_{{s = 1}}$$ of arithmetic sequences the covering function is w(x) = |{1 ≤ s ≤ k : x ≡ as (mod ns)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n0 of w(x) and the moduli n1, . . . , nk in such a system A. Here are three central results: (a) For each r=0,1, . . .,nk/(n0,nk)−1 there exists a J c {1, . . . , k−1} such that $${\sum\nolimits_{s \in J} {1/n_{s} = r/n_{k} } }$$. (b) If n1 ≤···≤nk−l <nk−l+1 =···=nk (0 < l < k), then for any positive integer r < nk/nk−l with r ≢ 0 (mod nk/(n0,nk)), the binomial coefficient $${\left( {\begin{array}{*{20}c} {l} \\ {r} \\ \end{array} } \right)}$$ can be written as the sum of some (not necessarily distinct) prime divisors of nk. (c) max(x∈ℤw(x) can be written in the form $${\sum\nolimits_{{\left( {s = 1} \right)}}^k {m_{s} /n_{s} } }$$ where m1, . . .,mk are positive integers.

Blocks with a cyclic defect group