2019/03/20 by Ivar Skau, Ivar Henning Skau, Skau, Ivar Henning +2 · 1 citation
Chemistry · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #History and advancements in chemistry #math.CO
paper · pdf · doi:10.48550/arxiv.1903.08379
arxiv created 2019/03/20 · openalex publication_date 2019/03/20 · arxiv updated 2019/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain partitioning process that gives rise to a sequence of sets of "nested" partitions. We prove that at stage m, the cardinality of the resulting set will equal the m-th order Bell number. This set-theoretic interpretation enables us to make a natural definition of higher order Stirling numbers and to study the combinatorics of these entities. The cardinality of the elements of the constructed "hyper partition" sets are explored.