We show that the Eulerian-Catalan numbers enumerate Dyck permutations. We provide two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analogue of the Chung-Feller theorem.

In this work we explore the p-adic valuation of Eulerian numbers. We construct a tree whose nodes contain information about the p-adic valuation of these numbers. Using this tree, and some classical results for Bernoulli numbers, we compute the exact p divisibility for the Eulerian numbers when the first variable lies in a congruence class and p satisfies some regularity properties.

We show that an Eulerian triangulation of the Klein bottle has chromatic number equal to six if and only if it contains a complete graph of order six, and it is 5-colorable, otherwise. As a consequence of our proof, we derive that every Eulerian triangulation of the Klein bottle with facewidth at least four is 5-colorable.

‎in this paper we introduce the concept of order difference interval graph $gamma_{odi}(g)$ of a group $g$‎. ‎it is a graph $gamma_{odi}(g)$ with $v(gamma_{odi}(g)) = g$ and two vertices $a$ and $b$ are adjacent in $gamma_{odi}(g)$ if and only if $o(b)-o(a) in [o(a)‎, ‎o(b)]$‎. ‎without loss of generality‎, ‎we assume that $o(a) leq o(b)$‎. ‎in this paper we obtain several properties of $gamma_...

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which have the property of specializing 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 distri...

The paper describes two algorithms for threading unknown, finite directed Eulerian mazes. Each of these algorithms is performed by a traveling robot whose control is a finite-state automaton. It is assumed that each vertex has a circular list of its outgoing edges. The items of this list are called exits. Each of the algorithms puts in one of the exits of each vertex a scan pebble. These pebble...

Nash-Williams’ well-balanced orientation theorem [11] is extended for orienting several graphs simultaneously. We prove that if G1, ..., Gk are pairwise edge-disjoint subgraphs of a graph G, then G has a well-balanced orientation ~ G such that the inherited orientations ~ Gi of Gi are well-balanced for all 1 ≤ i ≤ k. We also have new results about simultaneous well-balanced orientations of non-...

A transition system T of an Eulerian graph G is a family of partitions of the edges incident to each vertex of G into transitions i.e. subsets of size two. A circuit decomposition C of G is compatible with T if no pair of adjacent edges of G is both a transition of T and consecutive in a circuit of C. We give a conjectured characterization of when a 4-regular graph has a transition system which...

We present a general Lagrangian formalism that allows the treatment of vorticity. We give solutions for the rotational perturbations up to the third-order in a flat background universe. We show how the primordial vorticity affects the evolution of the density fluctuation in high-density regions. The study of the galaxy formation based on gravitational instability has not revealed the origin of ...

It is shown that the existence of an Euler path in a recursive graph is complete for the class DΣ 3 of all set differences of two Σ 3 sets. The same problem for highly recursive graphs as well as automatic graphs is shown to be Π 2 -complete. Moreover, the arithmetic level for bounding the number of ends in an automatic/recursive graph as well as computing the number of infinite paths in an aut...