The main result of this paper is the generalization and proof of a conjecture by Gould and Quaintance on the asymptotic behavior of certain sequences related to the Bell numbers. Thereafter we show some applications of the main theorem to statistics of partitions of a finite set S, i.e., collections B 1 , B 2 , . . . , B k of non-empty disjoint subsets of S such that k i=1 B i = S, as well as to certain classes of partitions of [n].
It is shown how arbitrary elements of the Weyl algebra can be represented by labeled plane trees using normal ordering. Several examples are treated and combinatorial aspects are discussed. Also, possible avenues for future research are described.
The purpose of this paper is to find an explicit formula and asymptotic estimate for the total number of sum of weighted records over set partitions of [n] in terms of Bell numbers. For that we study the generating function for the number of set partitions of [n] according to the statistic sum of weighted records.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.