Related papers: Cesaro's integral formula for the Bell numbers (co…
We define a quantity $c_m(n,k)$ as a generalization of the notion of the composition of the positive integer $n$ into $k$ parts. We proceed to derive some known properties of this quantity. In particular, we relate two partial Bell…
We establish explicit formulas for Bell numbers and graphical Stirling numbers of complete multipartite graphs, complete bipartite graphs with removed perfect matchings, and Mycielskian trees. For complete multipartite graphs…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
Exact rational partitions are presented for Bernoulli and Euler numbers as novel sums involving Faulhaber and Sali\'e coefficients.
An ordered set-partition (or preferential arrangement) of n labeled elements represents a single ``hierarchy''; these are enumerated by the ordered Bell numbers. In this note we determine the number of ``hierarchical orderings'' or…
We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial,…
A new explicit closed-form formula for the multivariate $(n, k)$th partial Bell polynomial $B_{n,k} (x_1, x_2, ..., x_{n - k + 1})$ is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily…
In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply…
In the note, the author discovers an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
A paper on ordinal partitions by Erd\H{o}s and Milner (1972) has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo-Fraenkel set theory. The work is part of a project on formalising the partition…
The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a…
A class of self-similar sets of entangled quantum states is introduced, for which a recursive definition is provided. These sets, the "Bell gems," are defined by the subsystem exchange symmetry characteristic of the Bell states. Each Bell…
Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives an explicit formula for Bernoulli numbers of even index. The formula contains a remarkable sequence of determinants.
In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also…
A novel general ready-to-use bit-error rate (BER) expression for one-dimensional constellations is developed. The BER analysis is performed for bit patterns that form a labeling. The number of patterns for equally spaced M-PAM…
We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.
We correct a 1957 combinatorial enumeration by the linguist J. Greenberg. The desired count, the Bell number B(25), supported using his Mass Comparison method for language classification. In 1987, he used this method to classify indigenous…
In the paper it is reported that Bell's correlation formula allows an Einstein local hidden variables explanation. The key is the application of Petis integration.
The Stirling number of the second kind $n\brace k$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls…
This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the…