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.

We present a method for the enumeration of restricted words over a finite alphabet. Restrictions are described through the inclusion or exclusion of suitable building blocks used to construct the words by concatenation. Our approach, which relies on the invert transform and its representation in terms of partial Bell polynomials, allows us to generalize and address in a systematic manner previo...

Abstract. We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobiński relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demons...

In this paper, we solve a general, four-parameter recurrence by both algebraic and combinatorial methods. The Bell numbers and some closely related sequences are solutions to the recurrence corresponding to particular choices of the parameters.

We describe a concrete construction of all pushout complements for two given morphisms f : A→ B, m : B→ D in the category of hypergraphs, valid also for the case where f ,m are non-injective. It is based on the generation of suitable equivalence relations. We also give a combinatorial interpretation and show how well-known coefficients from combinatorics, such as the Bell numbers, can be recove...

We investigate extremal functions exe(F, n) and exi(F, n) counting maximum numbers of edges and maximum numbers of vertex-edge incidences in simple hypergraphs H which have n vertices and do not contain a fixed hypergraph F ; the containment respects linear orderings of vertices. We determine both functions exactly if F has only distinct singleton edges or if F is one of the 55 hypergraphs with...

A 2-cover is a multiset of subsets of [n] := {1, 2, . . . , n} such that each element of [n] lies in exactly two of the subsets. A 2-cover is called proper if all of the subsets of distinct, and is called restricted if any two of them intersect in at most one element. In this paper we find asymptotic enumerations for the number of line graphs on n-labelled vertices and for 2-covers. We find tha...

We study a class of Padovan-like sequences that can be generated using special matrices of the third order. We show that terms of any sequence of this class can be expressed via Bell polynomials and their derivatives that use as arguments terms of another such sequence with smaller indexes. CAS Mathematica is used for cumbersome calculations and hypothesis testing.

This study uses the National Educational Longitudinal Survey of 1988 (NELS) to replicate both the analysis in The Bell Curve and that of several of its previous replications. We examine the relative importance of test scores and family background in predicting dropping out of high school, starting college, arrests, and out-of-wedlock fertility. Our results relax several arbitrary assumptions ma...

The Stirling number of the second kind {k} counts the number of ways to partition a set of n labeled balls into k non-empty unlabeled cells. We extend this problem and give a new statement of the r-Stirling numbers of the second kind and r-Bell numbers. We also introduce the r-mixed Stirling number of the second kind and r-mixed Bell numbers. As an application of our results we obtain a formula...