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.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)