We develop an extension of the classical Bell polynomials introducing the Laguerre-type version of this well-known mathematical tool. The Laguerre-type Bell polynomials are useful in order to compute the nth Laguerre-type derivatives of a composite function. Incidentally, we generalize a result considered by L. Carlitz in order to obtain explicit relationships between Bessel functions and gener...

Here presented are the definitions of (c)-Riordan arrays and (c)-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials. The characterization of (c)-Riordan arrays by means of the Aand Z-sequences is given, which corresponds to a horizontal construction of a (c)Riordan array rather than its definition approach through column generating functions. There exists...

The partial r-Bell polynomials generalize the classical partial Bell polynomials (coinciding with them when r = 0) by assigning a possibly different set of weights to the blocks containing the r smallest elements of a partition no two of which are allowed to belong to the same block. In this paper, we study the partial r-Bell polynomials from a combinatorial standpoint and derive several new fo...

Let S be a set with n elements. The set of all subsets of S has elements. By a partition of S we mean a disjoint set of nonempty subsets (called blocks) whose union is S. The set of all partitions of S has elements, where is the nth Bell number and where denotes the Stirling number of the second kind. For example, We have recurrences ([1,2,3]) and asymptotics where is defined by. If U and V are...

This entry provides formulae for counting the number of equivalence relations and partial equivalence relations over a finite carrier set with given cardinality. To count the number of equivalence relations, we provide bijections between equivalence relations and set partitions [4], and then transfer the main results of the two AFP entries, Cardinality of Set Partitions [1] and Spivey’s General...

We prove that if μ+ < λ = cf(λ) < μא0 for some regular μ > 2א0 , then there is no family of less than μא0 c-algebras of size λ which are jointly universal for c-algebras of size λ. On the other hand, it is consistent to have a cardinal λ ≥ א1 as large as desired and satisfying λ<λ = λ and 2λ + > λ++, while there are λ++ c-algebras of size λ+ that are jointly universal for calgebras of size λ+. ...

We introduce partial r-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula. Résumé Les r−polynômes de Bell dans des algèbres de Hopf combinatoires Nous définissons des polynômes r-Bell partiels dans trois algèbres de Hopf combinatoires. Nous prouvons une formule de factorisation po...

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with nested co...

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