In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Sti...

Although researchers have studied which factors influence the behavior of traditional search algorithms, currently not much is known about how domain properties influence the performance of real-time search algorithms. In this paper we demonstrate, both theoretically and experimentally, that Eulerian state spaces(a supersetof undirected state spaces)are very easy for some existing real-time sea...

A model of the gravitationally evolved dark matter distribution, in the Eulerian space, is developed. It is a simple extension of the excursion set model that is commonly used to estimate the mass function of collapsed dark matter haloes. In addition to describing the evolution of the dark matter itself, the model allows one to describe the evolution of the Eulerian space distribution of the ha...

Let G be an eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of G is an eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compa...

A simple characterization of the 3, 4, or 5-colorable Eulerian triangulations of the projective plane is given.

We give a method for computing the q-Eulerian distribution W (t, q) = w∈W t des(w) q l(w) as a rational function in t and q, where (W, S) is an arbitrary Coxeter system, l(w) is the length function in W , and des(w) is the number of simple reflections s ∈ S for which l(ws) < l(w). Using this we compute generating functions encompassing the q-Eulerian distributions of the classical infinite fami...

Eulerian numbers (and “Alternate Eulerian numbers”) are often interpreted as distributions of statistics defined over the Symmetric group. The main purpose of this paper is to define a way to represent permutations that provides some other combinatorial interpretations of these numbers. This representation uses a one-to-one correspondence between permutations and the so-called subexceedant func...

The combinatorial Hopf algebra on building sets BSet extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized DehnSomm...

The spatial dependence of Lagrangian displacement and velocity statistics is studied in the context of a data assimilating numerical model of the Gulf Mexico. In the active eddy region of the Western Gulf, a combination of Eulerian and Lagrangian measures are used to locate strongly hyperbolic regions of the flow. The statistics of the velocity field sampled by sets of drifters launched specifi...