We provide a probabilistic approach using renewal theory to derive some novel identities involving Eulerian numbers and uniform B-splines. The renewal perspective leads to a unified treatment for the normalized binomial coefficients and the normalized Eulerian numbers when studying their limits of sums, as well as their associated distributions - the binomial distributions (Bernstein polynomials) and the Irwin-Hall distributions (uniform B-splines). We further extend the probabilistic perspective to -Bernstein polynomials (Pólya-Eggenberger distributions) through conditional renewal processes, and derive new limits of various ways of summing for the two special numbers and associated distributions. The proposed probabilistic unification dramatically simplifies the proofs of some identities, which are far from obvious (such as for -Bernstein polynomials) or do not otherwise even appear promising (such as for Eulerian numbers).

The quasivariety of groupoids [Formula: see text] satisfying the implication [Formula: see text] generalises rectangular semigroups and central groupoids. We call them and find three combinatorial structures based upon arrays, matrices and graphs that are closely related. These generalise several groupoids of independent interest. The quasivariety generates the variety of all groupoids; they satisfy no nontrivial equations. We see some strong connections with isotopy, this being one of the classes of algebras (along with quasigroups) closed under isotopy. We investigate some constructions and show that a regular automorphism exists iff the groupoid is derived from a group via a Cayley graph construction.

Bousquet-Mélou and Petkovšek investigated the generating functions of multivariate linear recurrences with constant coefficients. We will give a reinterpretation of their results by means of division theorems for formal power series, which clarifies the structural background and provides short, conceptual proofs. In addition, extending the division to the context of differential operators, the case of recurrences with polynomial coefficients can be treated in an analogous way.

The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes-a purely combinatorial one and two geometric ones. It is shown, that most of the properties, which are known to be true for coloring complexes of graphs, break down in this more general setting, e.g., Cohen-Macaulayness and partitionability. Nevertheless, we are able to provide bounds for the [Formula: see text]- and [Formula: see text]-vectors of those complexes which yield new bounds on chromatic polynomials of hypergraphs. Moreover, though it is proven that the coloring complex of a hypergraph has a wedge decomposition, we provide an example showing that in general this decomposition is not homotopy equivalent to a wedge of spheres. In addition, we can completely characterize those hypergraphs whose coloring complex is connected.

This is a continuation of the study, begun by Ceccherini-Silberstein and Woess (2009) [5], of context-free pairs of groups and the related context-free graphs in the sense of Muller and Schupp (1985) [22]. The graphs under consideration are Schreier graphs of a subgroup of some finitely generated group, and context-freeness relates to a tree-like structure of those graphs. Instead of the cones of Muller and Schupp (1985) [22] (connected components resulting from deletion of finite balls with respect to the graph metric), a more general approach to context-free graphs is proposed via tree sets consisting of cuts of the graph, and associated structure trees. The existence of tree sets with certain "good" properties is studied. With a tree set, a natural context-free grammar is associated. These investigations of the structure of context free pairs, resp. graphs are then applied to study random walk asymptotics via complex analysis. In particular, a complete proof of the local limit theorem for return probabilities on any virtually free group is given, as well as on Schreier graphs of a finitely generated subgoup of a free group. This extends, respectively completes, the significant work of Lalley (1993, 2001) [18,20].

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) [Formula: see text] of a tree T of given order. While the trees that attain the lower bound are easily characterised, the trees with the largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order n is at most [Formula: see text] (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by Górska and Skupień on the number of maximal matchings (maximal with respect to set inclusion).

Let k, m, and r be three integers such that 2≤k≤m≤r. Let G be a 2r-regular, 2m-edge-connected graph of odd order. We obtain some sufficient conditions for G-v to contain a k-factor for all v∈V(G).

We study the distribution of triples of codewords of codes and ordered codes. Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (8) (2005) 2859-2866] used the triple distribution of a code to establish a bound on the number of codewords based on semidefinite programming. In the first part of this work, we generalize this approach for ordered codes. In the second part, we consider linear codes and linear ordered codes and present a MacWilliams-type identity for the triple distribution of their dual code. Based on the non-negativity of this linear transform, we establish a linear programming bound and conclude with a table of parameters for which this bound yields better results than the standard linear programming bound.

Various topological indices have been put forward in different studies from bio-chemistry to pure mathematics. Among them the Wiener index, the number of subtrees and the Randić index have received great attention from mathematicians. While studying the extremal problems regarding these indices among trees, one interesting phenomenon is that they share the same extremal tree structures. Much effort was devoted to the study of the correlations between these various indices. In this note we provide a common characteristic (the 'semi-regular' property) of these extremal structures with respect to the above mentioned indices, among trees with a given maximum degree. This observation leads to a more unified approach for characterizing these extremal structures. As an application/example, we illustrate the idea by studying the extremal trees regarding the sum of distances between all pairs of leaves of a tree, a new index, which recently appeared in phylogenetic tree reconstruction and the study of the neighborhood of trees.