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It is well known that the Bell numbers represent the total number ofpartitions 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 acertain 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 willequal the m-th order Bell number. This set-theoretic interpretation enables usto make a natural definition of higher order Stirling numbers and to study thecombinatorics of these entities. The cardinality of the elements of theconstructed "hyper partition" sets are explored.

Sets of iterated Partitions and the Bell iterated Exponential Integers