Geodesic
Enumeration of partitions by long rises, levels, and descents
Journal of integer sequences, Tome 12 (2009) no. 1
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 
@article{JIS_2009__12_1_a3,
     author = {Mansour,  Toufik and Munagi,  Augustine O.},
     title = {Enumeration of partitions by long rises, levels, and descents},
     journal = {Journal of integer sequences},
     year = {2009},
     volume = {12},
     number = {1},
     zbl = {1165.05308},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a3/}
}
TY  - JOUR
AU  - Mansour,  Toufik
AU  - Munagi,  Augustine O.
TI  - Enumeration of partitions by long rises, levels, and descents
JO  - Journal of integer sequences
PY  - 2009
VL  - 12
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a3/
LA  - en
ID  - JIS_2009__12_1_a3
ER  - 
%0 Journal Article
%A Mansour,  Toufik
%A Munagi,  Augustine O.
%T Enumeration of partitions by long rises, levels, and descents
%J Journal of integer sequences
%D 2009
%V 12
%N 1
%U http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a3/
%G en
%F JIS_2009__12_1_a3
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/