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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2023

Counting partitions by genus I. Genus 0 to 2

Résumé

The counting of partitions according to their genus is revisited. The case of genus 0non-crossing partitions-is well known. Our approach relies on two pillars : first a functional equation between generating functions, originally written in genus 0 and interpreted graphically by Cvitanovic, is generalized to higher genus; secondly, we show that all partitions may be reconstructed from the "(semi)-primitive" ones introduced by Cori and Hetyei. Explicit results for the generating functions of all types of partitions are obtained in genus 1 and 2. This gives a second order interpolation between expansions on ordinary or on free cumulants.
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Dates et versions

hal-04025762 , version 1 (13-03-2023)

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Jean-Bernard Zuber. Counting partitions by genus I. Genus 0 to 2. 2023. ⟨hal-04025762⟩
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