Recounting the Number of Rises, Levels, and Descents in Finite Set Partitions

Abstract A finite set partition is said to have a descent at i if it has a descent at i in its canonical representation as a restricted growth function (and likewise for level and rise). In this note, we provide direct combinatorial proofs as well as extensions of recent formulas for the total number of rises, levels, and descents in all the partitions of an n-set with a prescribed number of blocks. In addition, we supply direct proofs of formulas for the number of partitions having a fixed number of levels.