In this paper, we show new proofs of some important formulas by means of Liu’s expansion formula. Our results include a new proof of the identity for sums of two squares, a new proof of Gauss’s identity, a new proof of Euler’s identity, and a new proof of the identity for sums of four squares.

We explicitly evaluate the generating functions for joint distributions of pairs of the permutation statistics \(\text{inv}, {maj}\), and \({ch}\) over the symmetric group when both variables are set to \(-1\). We give a combinatorial proof by means of a sign-reversing involution that specializing the variables to \(-1\) in these bimahonian generating functions gives the number of two-colored permutations up to sign.

General methods for the construction of magic squares of any order have been searched for centuries. Several `standard strategies’ have been found for this purpose, such as the `knight movement’, or the construction of bordered magic squares, which played an important role in the development of general methods.

What we try to do here is to give a general and comprehensive approach to the construction of magic borders, capable of assuming methods produced in the past as particular cases. This general approach consists of a transformation of the problem of constructing magic borders to a simpler – almost trivial – form. In the first section, we give some definitions and notation. The second section consists of the exposition and proof of our method for the different cases that appear (Theorems 1 and 2). As an application of this method, in the third section we characterize magic borders of even order, giving therefore a first general result for bordered magic squares.

Although methods for the construction of bordered magic squares have always been presented as individual successful attempts to solve the problem, we will see that a common pattern underlies the fundamental mechanisms that lead to the construction of such squares. This approach provides techniques for constructing many magic bordered squares of any order, which is a first step to construct all of them, and finally know how many bordered squares are for any order. These may be the first elements of a general theory on bordered magic squares.

The main purpose of this paper is to define a pair of Konhauser matrix polynomials and obtain some properties, such as recurrence relations and matrix differential equations, for Konhauser matrix polynomials.

Studying expressions of the form \((f(z)D)^n\), where \(D = \frac{d}{dx}\) is the derivation operator, goes back to Scherk’s Ph.D. thesis in 1823. We show that this can be extended as
\(\sum{\gamma_{p;a}}(f^{(0)})^{a(0)+1}(f^{(1)})^{a(1)}\ldots (f^{(p-1)})^{a(p-1)}D^{p-\sum_i ia(i)},\) where the summation is taken over the \(p\)-tuples \((a_0, a_1, \ldots, a_{p-1})\), satisfying \(\sum_ia(i)=p-1 + ,\sum_iia(i) < p\), \(f^{(i)} = D^if\), and \(\gamma_{p;a}\) is the number of increasing trees on the vertex set \([0, p]\) having \(a(0) + 1\) leaves and having \(a(i)\) vertices with \(i\) children for \(0 < i < p\). Thus, previously known results about increasing trees lead us to some equalities containing coefficients \(\gamma_{p;a}\). In the sequel, we consider the expansion of \({(x^kD)}^p\) and coefficients appearing there, which are called generalized Stirling numbers by physicists. Some results about these coefficients and their inverses are discussed through bijective methods. Particularly, we introduce and use the notion of \((p,k)\)-forest in these arguments.

In this note, we determine the exact value for the second largest eigenvalue of the derangement graph, by deriving a formula for all the eigenvalues corresponding to the \(2\)-part partitions. This result is then used to obtain.

Since ancient times, mathematicians have considered geometrical objects with integral side lengths. We consider plane integral point sets \(P\), which are sets of \(n\) points in the plane with pairwise integral distances, where not all the points are collinear.

The largest occurring distance is called its diameter. Naturally, the question about the minimum possible diameter \(d(2, 7)\) of a plane integral point set consisting of \(7\) points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets, we prove a lower bound for \(d(2, n)\) achieving the known upper bound \(n^{c_2\log \log n }\) up to a constant in the exponent.
A famous question of Erdés asks for plane integral point sets with no \(3\) points on a line and no \(4\) points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by \(d(2,n)\). Recently \(d(2, 7) = 22270\) could be determined via an exhaustive search.

In this paper, we study invariant sequences by umbral method, and give some identities which are similar with the identities of Bernoulli numbers.

In this paper, we consider the total domination number, the restrained domination number, the total restrained domination number and the connected domination number of lexicographic product graphs.