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 (introduced by Poneti and Vajnovzski) and the second kinds. We present two other Bell number enumerated classes of subexcedant functions. Further, we give bijections on set partitions, in particular, an involution that interchanges the sets of merging blocks and successions. We use the bijections to enumerate the distribution of these statistics over set partitions and give some structural and enumeration results.