Counting partitions of a fixed genus

TL;DR

This paper proves that for any fixed genus, the generating function counting partitions of an n-element set into k blocks is algebraic, using a reduction to primitive partitions and a computer-assisted enumeration for genus 2.

Contribution

It introduces a novel algebraic approach to counting genus g partitions by reducing to primitive partitions and demonstrates this with genus 2 cases.

Findings

The generating function for fixed genus partitions is algebraic.

Primitive partitions of a given genus are finitely many.

Explicit genus 2 generating function was obtained using computational methods.

Abstract

We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus $2$ partitions, after identifying all genus $2$ primitive partitions, using a computer-assisted search.