Enumeration of partitions by rises, levels and descents

A descent in a permutation α 1 α 2 · α n is an index i for which α i > α i +1 . The number of descents in a permutation is a classical permutation statistic which was first studied by P. A. MacMahon almost a hundred years ago, and it still plays an important role in the study of permutations. Representing set partitions by equivalent canonical sequences of integers, we study this statistic among the set partitions, as well as the numbers of rises and levels . We enumerate set partitions with respect to these statistics by means of generating functions, and present some combinatorial proofs. Applications are obtained to new combinatorial results and previously-known ones. Introduction A descent in a permutation α = α 1 α 2 ··· α n is an index i for which α i > α i +1 . The number of descents in a permutation is a classical permutation statistic. This statistic was first studied by MacMahon, and it still plays an important role in the study of permutation statistics. In this paper we study the statistics of numbers of rises, levels and descents among set partitions expressed as canonical sequences, defined below.