Enumeration of partitions by long rises, levels, and descents

When the partitions of [n] = {1,2,⋯, 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.