Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.8

Set Partitions and Other Bell Number Enumerated Objects


Fufa Beyene
Department of Mathematics
Addis Ababa University
1176 Addis Ababa
Ethiopia

Jörgen Backelin
Department of Mathematics
Stockholm University
SE-106 91 Stockholm
Sweden

Roberto Mantaci
IRIF, Université de Paris
8 Place Aurélie Nemours
F-75013 Paris
France

Samuel A. Fufa
Department of Mathematics
Addis Ababa University
1176 Addis Ababa
Ethiopia

Abstract:

In this paper, we study classes of subexcedant functions enumerated by the Bell numbers and present bijections on set partitions. We present a set of permutations whose transposition arrays are the restricted growth functions, thus defining Bell permutations of the second kind. We describe a bijection between Bell permutations of the first (introduced by Poneti and Vajnovzski) and the second kinds. We present two other Bell number enumerated classes of subexcedant functions. Further, we give bijections on set partitions, in particular, an involution that interchanges the sets of merging blocks and successions. We use the bijections to enumerate the distribution of these statistics over set partitions and give some structural and enumeration results.


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(Concerned with sequences A008299 A026898 A056857 A259691.)


Received September 25 2022; revised versions received January 3 2023; January 9 2023. Published in Journal of Integer Sequences, January 17 2023.


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