Summary: When the partitions of $[n] = \{1, 2,\dots, n\}$ are identified with the restricted growth functions on $[n]$, under a known bijection, certain enumeration problems for classical word statistics are formulated for set partitions. In this paper we undertake the enumeration of partitions of $[n]$ with respect to the number of occurrences of rises, levels, and descents, of arbitrary integral length not exceeding $n$. This approach extends previously known cases. We obtain ordinary generating functions for the number of partitions with a specified number of occurrences of the three statistics. We also derive explicit formulas for the number of occurrences of each statistic among all partitions, besides other combinatorial results.