Related papers: A generalization of Eulerian numbers via rook plac…

For finite sums of non-negative powers of arithmetic progressions the generating functions (ordinary and exponential ones) for given powers are computed. This leads to a two parameter generalization of Stirling and Eulerian numbers. A…

We consider the joint distribution of the area and perimeter statistics on the set I_n of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived…

In this article we generalize the $q$-difference operator due to Carlitz in order to derive explicit sum formulae for several extensions of Stirling numbers of the second kind, including complete homogeneous symmetric functions,…

We show how some special production matrices may be used to define families of generalized Eulerian triangles. We furthermore show that these generalized Eulerian triangles are the coefficient arrays of polynomials which are the moments of…

Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor…

In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic…

In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore…

We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the $n$th term of the sequence as being the total weight of the integer $n$ written in base $k$.…

We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their…

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…

Euler systems are certain compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou…

Let $S$ be a set of $r$ red points and $b=r+2d$ blue points in general position in the plane, with $d\geq 0$. A line $\ell$ determined by them is said to be balanced if in each open half-plane bounded by $\ell$ the difference between the…

Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and…

A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only…

Consider n unit intervals, say [1,2], [3,4], ..., [2n-1,2n]. Identify their endpoints in pairs at random, with all (2n-1)!! = (2n-1) (2n-3) ... 3 1 pairings being equally likely. The result is a collection of cycles of various lengths, and…

In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group $G$. When $G$ is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley…

Consider a string of $n$ positions, i.e. a discrete string of length $n$. Units of length $k$ are placed at random on this string in such a way that they do not overlap, and as often as possible, i.e. until all spacings between neighboring…

The Eulerian number A(n,k) counts permutations of n symbols with exactly k descents. Motivated by problems in cryptography, several authors have studied the proportion of permutations whose number of descents lies in a fixed congruence…

In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find…

The goal of this short paper to advertise the method of gauge transformations (aka holographic reduction, reparametrization) that is well-known in statistical physics and computer science, but less known in combinatorics. As an application…