Journal of Integer Sequences, Vol. 12 (2009), Article 09.1.8

Enumeration of Partitions by Long Rises, Levels, and Descents

Toufik Mansour
Department of Mathematics
Haifa University
31905 Haifa

Augustine O. Munagi
The John Knopfmacher Centre for Applicable Analysis and Number Theory
School of Mathematics
University of the Witwatersrand
Johannesburg 2050
South Africa


When the partitions of [n] = {1, 2, ... , 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.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000110 A052889 A105479 A105480 A105481 A105483 A105484 A105485 and A105486.)

Received July 5 2008; revised version received January 2 2009. Published in Journal of Integer Sequences, January 3 2009.

Return to Journal of Integer Sequences home page