Definition. A set partition is a partition of the set [n] for some n into any number of nonempty sets, where the order within the partition is irrelevant. We call these sets blocks of the partition. The number of set partitions of [n] is the Bell number Bn. Often, when writing specific set partitions, we will write the partition [n] = S1 ∪ · · · ∪ Sk as S1/ · · · /Sk, where the Si are in increa...

The present paper considers a kind of combinatorial series and its allied reciprocal relations which are determined by discrete multi-fold convolutions. Furthermore, their various formal and analytic expressions in explicit forms are obtained. Constructive applications to some well-known sequences such as the Bell numbers, the Fibonacci numbers, the Stirling numbers and some others given by int...

We have considered a two-particle Bell experiment to visualize the conflict between rotational invariance of physical laws and specific local realistic theory. The experiment is reproducible by a local realistic theory which is obtained in a twosetting Bell experiment. It turns out that the generalized Bell inequality [J. Phys. A: Math. Theor. 40, 13101 (2007)] which is derived under the assump...

The Encyclopedia of Integer Sequences includes some sequences that are connected with the Bell numbers and that have a particular combinatorial meaning. In this article, we find a general meaning for framing sequences, including the above mentioned ones. Furthermore, by using Laguerre-type derivatives, we derive the Laguerre-type Bell numbers of higher order, showing, as a by-product, that it i...

Cardinalities and lattice structures which are attainable by semilattices of ordered compactifications of completely regular ordered spaces are examined. Visliseni and Flachsmeyer have shown that every infinite cardinal is attainable as the cardinality of a semilattice of compactifications of a Tychonoff space. Among the finite cardinals, however, only the Bell numbers are attainable as cardina...

The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers both kinds associated with hyperharmonic also Bernolli, Euler, Bell Fubini polynomials.

Recently, Kim–Kim introduced the [Formula: see text]-umbral calculus, in which text]-Sheffer sequences occupy central position. In this paper, we introduce fully degenerate Bell and Dowling polynomials, investigate some properties identities relating to those polynomials with help of calculus. Here, note that poynomials are, respectively, versions latters are natural extension Whitney numbers s...