Counting partitions by genus. I. Genus 0 to 2
Combinatorics
2023-05-04 v2 Mathematical Physics
math.MP
Abstract
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
Cite
@article{arxiv.2303.05875,
title = {Counting partitions by genus. I. Genus 0 to 2},
author = {Jean-Bernard Zuber},
journal= {arXiv preprint arXiv:2303.05875},
year = {2023}
}
Comments
23 pages, 15 figures