Geodesic
Enumeration of partitions by long rises, levels, and descents
Journal of integer sequences, Tome 12 (2009) no. 1.

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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.
Classification : 05A05, 05A15
Keywords: set partition, generating function, recurrence relation, t-rise, t-level, t-descent 
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     author = {Mansour, Toufik and Munagi, Augustine O.},
     title = {Enumeration of partitions by long rises, levels, and descents},
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     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a3/}
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Mansour, Toufik; Munagi, Augustine O. Enumeration of partitions by long rises, levels, and descents. Journal of integer sequences, Tome 12 (2009) no. 1. http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a3/