Rise, Fall and Level Statistics on r-Jacobi-Stirling Set Partitions

In this paper, we consider sequential representations of the recently introduced r-Jacobi-Stirling set partitions (denoted by P(n, k)) and study various statistics on these representations.We compute an explicit formula for the generating function which counts members ofP(n, k) where k and r are fixed according to these statistics in the case of levels, descents and ascents. In each case, we use a more-or-less uniform strategy which also yields the distribution of the statistic on those members ofP(n, k) ending in a certain letter. Finally, we give explicit formulas for the total number of levels, descents and ascents within all of the members of P(n, k), providing both algebraic and combinatorial proofs.