Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6

Counting Partitions by Genus: a Compendium of Results

Robert Coquereaux
Aix Marseille Université
Université de Toulon, CNRS, CPT
F-13284 Marseille Cedex 09

Jean-Bernard Zuber
Sorbonne Université, CNRS
Laboratoire de Physique Théorique et Hautes Energies, LPTHE
F-75252 Paris


We study the enumeration of set partitions, according to their length, number of parts, cyclic type, and genus. We introduce genus-dependent Bell, Stirling numbers, and Faà di Bruno coefficients. Besides attempting to summarize what is already known on the subject, we obtain new generic results (in particular for partitions into two parts, for arbitrary genus), and present computer generated new data extending the number of terms known for sequences or families of such coefficients; this also leads to new conjectures.

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(Concerned with sequences A000108 A000110 A000292 A000296 A000581 A001263 A001287 A001700 A002411 A002450 A002802 A005043 A008277 A008299 A025035 A025036 A025037 A025038 A025039 A035319 A059260 A103371 A108263 A134991 A144431 A185259 A245551 A275514 A297178 A297179 A340556.)

Received June 29 2023; revised versions received July 1 2023; January 15 2024; January 31 2024; February 2 2024; February 4 2024. Published in Journal of Integer Sequences, February 5 2024.

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