Related papers: Bernoulli Partitions

It is known that the ordered Bell numbers count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$. In this paper, we introduce the deranged Bell numbers that count the total number of deranged partitions of $[n]$. We first study…

In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind.

The Bernoulli numbers b_0,b_1,b_2,.... of the second kind are defined by \sum_{n=0}^\infty b_nt^n=\frac{t}{\log(1+t)}. In this paper, we give an explicit formula for the sum \sum_{j_1+j_2+...+j_N=n,…

The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n >0$ is the antiderivative of $b_{n-1}(x)$ subject to $\int_0^1 b_n(x) dx = 0$. We offer a related…

In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary…

We evaluate in closed form series of the type $\sum u(n) R(n)$, where $(u(n))_n$ is a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is: $$ \sum_{n \geq 1} (-1)^{s_2(n)}…

The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n,…

In this paper we prove that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for $ n $ large enough, the two…

We say that a random integer variable $X$ is monotone if the modulus of the characteristic function of $X$ is decreasing on $[0,\pi]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random…

We introduce a novel generalization of deranged Bell numbers by defining the partial deranged Bell numbers $w_{n,r}$, which count the number of set partitions of $\left[ n\right] $ with exactly $r$ fixed blocks, while the remaining blocks…

Given an integer base $b\geq 2$, a number $\rho\geq 1$ of colors, and a finite sequence $\Lambda=(\lambda_1,\ldots,\lambda_\rho)$ of positive integers, we introduce the concept of a $\Lambda$-restricted $\rho$-colored $b$-ary partition of…

We study semantic and syntactic properties of spherical orders and their elementary theories, including finite and dense orders and their theories. It is shown that theories of dense $n$-spherical orders are countably categorical and…

A rank $n$ generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. In this paper we classify these groups in terms of their separability…

A partition is finitary if all its blocks are finite. For a cardinal $\mathfrak{a}$ and a natural number $n$, let $\mathrm{fin}(\mathfrak{a})$ and $\mathscr{B}_{n}(\mathfrak{a})$ be the cardinalities of the set of finite subsets and the set…

Let i.i.d. symmetric Bernoulli random variables be associated to the edges of a binary tree having n levels. To any leaf of the tree, we associate the sum of variables along the path connecting the leaf with the tree root. Let M_n denote…

Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as…

We introduce new entanglement monotones which generalize, to the case of many parties, those which give rise to the majorization-based partial ordering of bipartite states' entanglement. We give some examples of restrictions they impose on…

Let X be a locally compact Abelian group. We consider linear forms of independent random variables with values in X. In doing so, one of the coefficients of the linear forms is a random variable with a Bernoulli distribution. For some…

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability…

In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a…