  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.
