  In this paper, we study classes of subexcedant functions enumerated by the
Bell numbers and present bijections on set partitions. We present a set of
permutations whose transposition arrays are the restricted growth functions,
thus defining Bell permutations of the second kind. We describe a bijection
between Bell permutations of the first kind (introduced by Ponti and
Vajnovzski) and the second kind. We present two other Bell number enumerated
classes of subexcedant functions. Further, we present bijections on set
partitions, in particular, an involution that interchanges the set of merging
blocks and the set of successions. We use the bijections to enumerate the
distribution of these statistics over the set of set partitions, and also give
some enumeration results.
