Related papers: Cesaro's integral formula for the Bell numbers (co…

We examine and clarify in detail the contributions of Yoshisuke Matsunaga (1694?--1744) to the computation of Bell numbers in the eighteenth century (in the Edo period), providing modern perspectives to some unknown materials that are by…

A new family of generalized Pell numbers was recently introduced and studied by Br\'od \cite{Dorota}. These number possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can…

We study violations of n particle Bell inequalities (as developed by Mermin and Klyshko) under the assumption that suitable partial transposes of the density operator are positive. If all transposes with respect to a partition of the system…

Let B_n denote the nth Bell number. We use well-known recursive expressions for B_n to give a generalizing recursion that can be used to prove Touchard's congruence.

For a given sequence of weights (non-negative numbers), we consider partitions of the positive integer n. Each n-partition is selected uniformly at random from the set of all such partitions. Under a classical scheme of assumptions on the…

We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…

In a recent series of papers Wiseman, Cavalcanti, and Rieffel have outlined and contrasted two different views about what we now call Bell's theorem. They also assert that Bell presented these two different versions at different times. This…

We revisit a formula for the number of plane partitions due to Almkvist. Using the circle method, we provide modifications to his formula along with estimates of the errors. We show that the improved formula continues to be an asymptotic…

A decomposition form is introduced in this report to establish a criterion for the bi-partite separability of Bell diagonal states. A such criterion takes a quadratic form of the coefficients of a given Bell diagonal states and can be…

Consider the set $\{1,2,\ldots,3n\}$. We are interested in the number of partitions of this set into subsets of three elements each, where the sum of two of them equals the third. We give some criteria such a partition has to fulfill, which…

To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add…

In studying the enumerative theory of super characters' of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions have simple closed…

Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations…

Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, which was the subject of an earlier paper, $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, can be…

Beck introduced two partition statistics $NT(r,m,n)$ and $M_{\omega}(r,m,n)$,which denote the total number of parts in the partition of $n$ with rank congruent to $r$ modulo $m$ and the total number of ones in the partition of $n$ with…

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if…

The aim of this paper is to introduce truncated degenerate Bell polynomials and numbers and to investigate some of their properties. In more detail, we obtain explicit expressions, identities involving other special polynomials, integral…

The ${\mathbb B}_n^{(k)}$ poly-Bernoulli numbers --- a natural generalization of classical Bernoulli numbers ($B_n={\mathbb B}_n^{(1)}$) --- were introduced by Kaneko in 1997. When the parameter $k$ is negative then ${\mathbb B}_n^{(k)}$ is…

In this paper we use Faa di Bruno's formula to associate Bell polynomial values to differential equations of the form $y^{\prime}=f(y)$. That is, we use partial Bell polynomials to represent the solution of such an equation and use the…