Extensions of Set Partitions and Permutations

Caicedo, Jhon B.; Moll, Victor H.; Ramírez, José L.; Villamizar, Diego

doi:10.37236/8223

Cited by 14 publications

(13 citation statements)

References 31 publications

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“…If we require each block/cycle/ordered block (not only the non-distinguished ones) to contain at least s elements, then we call them n, m, r , D A ≥s, * n, m, r , DL ≥ s, * n, m, r , respectively. Alternative s-associated r -Bell numbers appear in two recent papers [1,3] (the former one allows a more general restriction on the cardinality of the blocks), while in [19,24], the authors investigated alternative 2-associated and alternative s-associated r -factorials under the names of r -derangement and generalized r -derangement numbers.…”

Section: Alternative Definitionsmentioning

confidence: 99%

“…Again, B ≥ 1 n, r = B n, r . After the completion of this article, we have learnt about two very recent papers [1,3] which deal with associated r -Bell numbers, but in an alternative way, see the closing section of the present paper about alternative definitions. Now, we turn our attention to the "Bell-like numbers of the first kind", the permutational variants of the above numbers.…”

Section: Introductionmentioning

confidence: 99%

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Associated r-Dowling numbers and some relatives

Gyimesi¹,

Nyul²

2021

Comptes Rendus. Mathématique

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In this paper, we introduce a new generalization of Bell numbers, the s-associated r-Dowling numbers by combining two investigational directions. Here, r distinguished elements have to be in distinct blocks, some elements are coloured according to a colouring rule, and the cardinality of certain blocks is bounded from below by s. Along with them, we define some relatives, the s-associated r-Dowling factorials and the s-associated r-Dowling-Lah numbers, when the underlying set is decomposed into cycles or ordered blocks. The study of these numbers is based on their combinatorial meaning, and the exponential generating functions of their sequences derived from the so-called r-compositional formula.

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Section: Alternative Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Associated r-Dowling numbers and some relatives

Gyimesi¹,

Nyul²

2021

Comptes Rendus. Mathématique

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“…The r-version of special numbers is a common natural extension in enumerative combinatorics, see, for example, the r-Stirling numbers [4], r-Bell numbers [15], r-Fubini numbers [5], r-derangement numbers [19].…”

Section: The R-deranged Bell Numbermentioning

confidence: 99%

“…Most of the previous works focused on the some restrictions and generalizations of Stirling number of second kind to introduce new classes of ordered Bell number (see [5] and the references given there).…”

Section: Introductionmentioning

confidence: 99%

The deranged Bell numbers

Belbachir,

Djemmada,

Németh

2021

Preprint

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It is known that the ordered Bell numbers count all the ordered partitions of the set [n] = {1, 2, . . . , n}. In this paper, we introduce the deranged Bell numbers that count the total number of deranged partitions of [n]. We first study the classical properties of these numbers (generating function, explicit formula, convolutions, etc.), we then present an asymptotic behavior of the deranged Bell numbers. Finally, we give some brief results for their r-versions.

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“…Finally, we mention briefly a few more variants of Fubini numbers and polynomials without going into details, because they lie beyond the scope of this paper. We can prescribe certain restrictions on the cardinalities of the blocks, such restricted and associated Fubini numbers can be found in [5,10]. Fubini type polynomials related to Whitney and r-Whitney numbers appear in [4,15].…”

Section: Introductionmentioning

confidence: 99%