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arXiv · 1903.08379

Sets of iterated Partitions and the Bell iterated Exponential Integers

Ivar Henning Skau·Kai Forsberg Kristensen
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Abstract

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.

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BibTeXRIS

Ivar Henning Skau, Kai Forsberg Kristensen. 2019-03-20. Sets of iterated Partitions and the Bell iterated Exponential Integers. https://arxiv.org/abs/1903.08379

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