We give three proofs for the following symmetrical identity involving binomial coefficients ( n m ) and Eulerian numbers 〈 n m 〉 : ∑ k ( a + b k ) 〈 k a − 1 〉 = ∑ k ( a + b k ) 〈 k b − 1 〉 for any positive integers a and b (where we take 〈 0 0 〉 = 0). We also show how this fits into a family of similar (but more complicated) identities for Eulerian numbers.

We introduce a new array of type D Eulerian numbers, different from that studied by Brenti, Chow and Hyatt. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability distributions whose moments are Eulerian polynomials of type A, B and D.

We present an infinite series of n-state Eulerian automata whose reset words have length at least (n − 3)/2. This improves the current lower bound on the length of shortest reset words in Eulerian automata. We conjecture that (n − 3)/2 also forms an upper bound for this class and we experimentally verify it for small automata by an exhaustive computation.

The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found.

Atmospheric dispersion is a topic of great importance especially in relation to pollutant transport. Two different approaches, known as the Eulerian and the Lagrangian frameworks, are used to describe this process. In the Eulerian framework, statistical properties are calculated in a fixed reference frame. This approach is most commonly used in field experiments as well as in laboratory experim...

Based Based on on two two inversion inversion formulas formulas for for enumerating enumerating words words in in the the free free monoid monoid by by adjacencies, adjacencies, we we present present a a new new approach approach to to a a class class of of permutation permutation problems problems having having Eulerian-type Eulerian-type generating generating functions. functions. We We also ...

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index a...

In this paper, we study some properties of Whitney numbers of Dowling lattices and related polynomials. We answer the following question: there is a relation between Stirling and Eulerian polynomials. Can we find a new relation between Dowling polynomials and other polynomials generalizing Eulerian polynomials? In addition, some congruences for the Dowling numbers are given. Mathematics Subject...