Counting partitions by genus. I. Genus 0 to 2

The counting of partitions according to their genus is revisited. The case of genus 0 -- non-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