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In state-of-the-art, it has proven that multi-view ensemble learning performs better than classical machine learning algorithms, with the optimized setting of views (subsets of features) usually. Obtaining the appropriate number of views for a given dataset is a complex problem in multi-view ensemble learning (MEL). The finding of the total number of possible views is an NP-hard problem, i.e., equivalent to Bell number. Moreover, the complexity of multi-view learning increases over a higher number of views of the dataset. Therefore, it is highly required to consider a smaller number of views with higher accuracy for optimal performance of MEL. In this work, MEL using Multi-Objective Particle Swarm Optimization (MEL-MOPSO) method has been proposed. The two objectives (number of views of the data and classification accuracy of MEL) have considered where the trade-off between objectives has been performed while searching for an optimal solution using Particle Swarm Optimization (PSO) in the process of multiobjective optimization. The experiments have been done over sixteen high-dimensional datasets using four state-of-art view construction methods. The individual views of the dataset has been utilized to learn through a support vector machine algorithm. The quantitative and non-parametric statistical analyses show that the proposed method has performed effectively and efficiently.

The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.

In many applications, like database management systems , is very useful to have an expression to compute the cardinality of cartesian product of k sets without repeated elements; we designate this problem as T ( k ) . The value of | T ( k ) | is upper-bounded by the multiplication of cardinalities of the sets. As long as we have searched, it has not been reported a general expression to compute T ( k ) using cardinalities of the intersections of sets, this is the main topic of this paper. Given three sets with indices { 0 , 1 , 2 } , C i is the cardinality of one set, C i , j ( i j ) and C i , j , l ( i j l ) are respectively the cardinalities of the intersections of 2 and 3 sets, then the searched formulas for T ( k ) are: T ( 1 ) = C 0 ; T ( 2 ) = C 0 C 1 - C 0 , 1 ; T ( 3 ) = C 0 C 1 C 2 - ( C 0 , 1 C 2 + C 0 , 2 C 1 + C 1 , 2 C 0 ) + 2 C 0 , 1 , 2 . In this paper, we prove formulas for computing T ( k ) and its specialization when a set is contained in the next sets. For this purpose, we will use concepts like partitions of the integer k in v parts, Bell numbers, Stirling numbers of the first kind and Stirling numbers of the second kind. Additionally, we present a complexity analysis for the computation of T ( k ) .

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 representations, a Dobinski-like formula and expressions of the generating function in terms of differential operators and the linear incomplete gamma function. In addition, we introduce truncated degenerate modified Bell polynomials and numbers and obtain similar results for those polynomials. As an application of our results, we show that the truncated degenerate Bell numbers can be expressed as a finite sum involving moments of a beta random variable with certain parameters.

The Set Partitioning Problem (SPP) aims to obtain non-empty disjoint subsets of objects such that their union equals the whole set of objects, and the partition meets some prespecified criteria. The ubiquity of SPP is impressive, given that it has a lot of theoretical and practical motivations. In the theoretical side, the study of the SPP is closely related to Bell numbers, Stirling numbers of the second kind, integer partitions, Eulerian numbers, Restricted Growth Strings (RGS), factoradic number system, power calculations, etc. In the practical side, SPP is intimately related to classification problems, clustering problems, reduction of dimensionality problems, and so on. In this work, three representations for instances of SPP are presented, these representations use: Restricted Growth Strings (RGS), factoradic number system, and a number system with a fixed base. Two cases for these representations will be presented: where the number of subsets is unbounded (i.e. the number of subsets can be the number of objects); and where the number of subsets is less than the number of objects. Bidirectional mappings between these three representations will be introduced, also the mapping among these three representations and the power of a base is defined. Given, that these three representations can be used to solve instances of SPP using exact, greedy, and metaheuristic algorithms, that require to do small changes to one possible solution and/or recombination of two possible solutions, definitions of mutation and recombination operators for the three representations will be shown. In order to motivate the use of the three representations for the solution of particular instances of SPP, it was decided to present their application to solve an instance of a set partition of integers problem (SPIP) using a simple genetic algorithm.

This paper aims to construct a new family of numbers and polynomials which are related to the Bell numbers and polynomials by means of the confluent hypergeometric function. We give various properties of these numbers and polynomials (generating functions, explicit formulas, integral representations, recurrence relations, probabilistic representation,...). We also derive some combinatorial sums including the generalized Bernoulli polynomials, lower incomplete gamma function, generalized Bell polynomials. Finally, by applying Cauchy formula for repeated integration, we introduce poly-Bell numbers and polynomials., 23 pages

Canonical expressions are representative of implicative propositions upto renaming of variables. In this paper we explore, using a Monte-Carlo approach, the model of canonical expressions in order to confirm the paradox that says that asymptotically almost all classical theorems are intuitionistic.

Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

The central factorial numbers of the second kind appear in the expansion of powers of x in terms of the central factorial sequence. In this paper, we introduce the central Bell numbers and polynomials associated with those central factorial numbers of the second kind and investigate some identities and properties of them.

Recently, the partially degenerate Bell numbers and polynomials were introduced as a degenerate version of Bell numbers and polynomials. In this paper, as a further degeneration of them, we study fully degenerate Bell numbers and polynomials. Among other things, we derive various expressions for the fully degenerate Bell numbers and polynomials.

In the present study, we introduced general a subclass of bi-univalent functions by using the Bell numbers and $q-$Srivastava Attiya operator. Also, we investigate coefficient estimates and famous Fekete-Szegö inequality for functions belonging to this interesting class.

Balogh, Bollobas and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers. We prove that every hereditary class that lies above the Bell numbers and has finite distinguishing number contains a boundary class for well-quasi-ordering. This means that any such hereditary class which in addition is defined by finitely many minimal forbidden induced subgraphs must contain an infinite antichain. As all hereditary classes below the Bell numbers are well-quasi-ordered, our results complete the answer to the question of well-quasi-ordering for hereditary classes with finite distinguishing number. We also show that the decision procedure of Atminas, Collins, Foniok and Lozin to decide the Bell number (and which now also decides well-quasi-ordering for classes of finite distinguishing number) has run time bounded by an explicit (quadruple exponential) function of the order of the largest minimal forbidden induced subgraph of the class., 21 pages, 1 figure

Let $\begin{array}{} \mathcal{S}^*_B \end{array}$ be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Bounds on the first three consecutive higher-order Schwarzian derivatives for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ are investigated.

The primary motivation of the paper is to study the generalized distribution for analytic function classes associated with error functions and Bell numbers. Further, the authors estimate the bounds for probability distribution series by means of q-difference operator using subordination principle.

In this paper, some properties for Tauber's generalized Stirling and Lah numbers are obtained including other forms of recurrence relations, orthogonality and inverse relations, rational generating function and explicit formual in symmetric function form. Moreover, a new explicit formula is derived, which is analogous to the Qi formula.

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established.Â

Let G be a graph on n vertices with bounded treewidth. We use $$f_k(G)$$ to denote the number of spanning forests of G with k components. Given a tree decomposition of width at most p of G, we present an algorithm to compute $$f_k(G)$$ for $$k = 1,2, \cdots , n$$ . The running time of our algorithm is $$O((B(p+1))^3pn^3)$$ , where $$B(p+1)$$ is the $$(p+1)$$ -th Bell number.

The sharp lower and upper estimates on the second- and third-order Hermitian–Toeplitz determinants for the classes of starlike functions associated with the modified sigmoid function and a related function, whose Taylor coefficients are the Bell numbers, are investigated. Further, the third and fourth Hankel determinants for these classes are also estimated.

Given a ring $$T_n\ (n\geqslant 2)$$ T n ( n ⩾ 2 ) of lower triangular $$n\times n$$ n × n matrices with entries from an arbitrary field F, we completely classify the orbits of free cyclic submodules of $$^2T_n$$ 2 T n under the action of the general linear group $$GL_2(T_n)$$ G L 2 ( T n ) . Interestingly, the total number of such orbits is found to be equal to the Bell number $$B_n$$ B n . A representative of each orbit is also given.

The nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

In this chapter, we summarize some basic theory concerning permutations, multilinear algebra, set partitions, and diagrams for usage throughout the later part of the book. We start with the notion of permutations and then discuss the tensor product (T-product), vec operator, and commutation matrices. A transformation of multiple T-products of vectors into a given order is also considered. In the section on symmetrization and multilinear algebra, we introduce the symmetrizer in the framework of multilinear algebra of Cartesian tensors. In connection with the symmetric subspace of tensors we construct linear operators in terms of matrices for the elimination of identical entries from a tensor as well as the duplication, triplication, quadruplication, etc. of tensors of distinct entries. The Partitions and Diagrams section includes the inclusive and exclusive method of extending partitions to derive all partitions of a finite set. The use of Bell polynomials and Bell numbers for obtaining the partitions is outlined. Partitions with lattice structure are considered mainly to discuss indecomposable partitions and diagrams. A discussion of the particular cases of diagrams, such as closed diagrams without loops and closed diagrams with arms and no loops, concludes this chapter.