相关论文: Sylvester's Double Sums: the general case
In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson's conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson's conjecture to the higher order integral polynomial case. However, they did not…
With every family of finitely many subsets of a finite-dimensional vector space over the Galois-field with two elements we associate a cyclic transversal polytope. It turns out that those polytopes generalize several well-known polytopes…
We use the rational tableaux introduced by Stembridge to give a bideterminant basis for a normal reductive monoid and for its variety of noninvertible elements. We also obtain a bideterminant basis for the full coordinate ring of the…
In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent…
In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In…
Mutual-visibility sets were motivated by visibility in distributed systems and social networks, and intertwine with several classical mathematical areas. Monotone properties of the variety of mutual-visibility sets, and restrictions of such…
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
We study the monodromy groups of compositions of two indecomposable polynomials. In particular, we show that such monodromy groups either fulfill a certain "largeness" property, or are in an explicit list of exceptions. Such largeness…
We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by…
A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of…
The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results of Sylvester (1851),…
Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.
We consider some known and some new properties of the family of polynomials introduced by Ted Suffridge in 1969. We begin by giving a brief overview of their extremal properties in classic and more recent work. We also give a compact form…
The Sylvester-Gallai theorem states that for a finite set of points in the plane, if every line determined by any two of these points also contains a third, then the set is necessarily made of collinear points. In this paper, we first…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double…
This is an expository paper aiming to introduce Zilber's Exponential Closedness conjecture to a general audience. Exponential Closedness predicts when (systems of) equations involving addition, multiplication, and exponentiation have…
In this paper, we investigate two methods to express the natural powers of $2$ as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients…
This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the…
Fix a natural $\alpha$. Let $n\ge \alpha$ be an integer. Consider the symmetric group $S_{\alpha+n}$ and its subgroup $S_n$. We consider the group algebra of $S_{\alpha+n}$ and its subalgebra $\mathbb{O}[\alpha;n]$ consisting of…