Genus Permutations and Genus Partitions

TL;DR

This work develops a genus-resolved framework for counting permutations and set partitions, showing that after a suitable variable change the generating series become rational with poles at ramification points. It provides a closed formula for the genus-$g$ generating function of permutations on the transformed variable $X(y)$ for all genera, and explicit results for partitions up to genus $2$, with Laplace-transform representations and cylinder generalisations. The results connect to Free Probability and Topological Recursion, and include concrete cylinder and multi-boundary extensions, offering practical residue-based methods to compute high-genus coefficients and to prove conjectures about genus partitions. Overall, the paper supplies explicit, computable tools for higher-genus combinatorics of maps and partitions, with implications for moduli space Euler characteristics and related enumerative structures.

Abstract

For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After a variable transformation, the generating series are rational functions with poles located at the ramification points in the new variable. The generating series for any genus is given explicitly for permutations and up to genus 2 for set partitions. Extending the topological structure not just by the genus but also by adding more boundaries, we derive the generating series of non-crossing partitions on the cylinder from known results of non-crossing permutations on the cylinder. Most, but not all, outcomes of this article are special cases of already known results, however they are not represented in this way in the literature, which however seems to be the canonical way. To make the article as accessible as possible, we avoid going into details into the explicit connections to Topological Recursion and Free Probability Theory, where the original motivation came from.

Figures (4)

• Figure 1: Two permutations $\sigma_1,\sigma_2\in \mathcal{S}_3$ with $\sigma_1=(1,2,3)$ and $\sigma_2=(1,3,2)$ are drawn together with its embedding on the corresponding Riemann surface. The permutation $\sigma_1$ is of genus $g=0$ and the permutation $\sigma_2$ of genus $g=1$. The orientation indicated by the arrows matters and the enclosed area is to the right of the orientation.
• Figure 2: Two partitions $\lambda,\lambda'\in P(4)$ with $\lambda=\{(1,2),(3,4)\}$ and $\lambda'=\{(1,3),(2,4)\}$ are drawn together with their embedding on the corresponding Riemann surfaces. The partition $\lambda$ is of genus $g=0$ and the partition $\lambda'$ of genus $g=1$. There is no orientation, the blocks of $\lambda,\lambda'$ are sets.
• Figure 3: On the left, we have the boundary permutation $\tau\in \mathcal{S}_{8}$ and the permutation $\sigma\in \mathcal{S}_{8}$ of genus $g=0$ of the form $\tau=((1,2,3,4,5),(1',2',3'))$ and $\sigma=((1,3'),(2,3,5,1',2'),(4))$. On the right, we have a partitioned permutation, where the associated weight is of the form $\kappa_1\kappa_2\kappa_{1,2}$, where $\kappa_1$ comes from the cycle $(1')$, $\kappa_2$ from $(3,4)$ and $\kappa_{1,2}$ from $(2'),(1,2)$ connected non-trivially.
• Figure 4: The permutations above show how one can construct the case (3) from the case (1) by sending $\kappa_{i,j}\mapsto i\cdot j\cdot \kappa_{i+j}$ since there are $i\cdot j$ possibilities to generate cycles of length $i+j$ where all the other trivial cycles are the same. The partitions below show that from the case (3) just one partition can be generated $\kappa_{i,j}\mapsto \kappa_{i+j}$. All $i\cdot j$ permutations above are in the same equivalence class as a partition.

Theorems & Definitions (22)

• Example 2.1
• Remark 2.2
• Remark 2.3
• Theorem 3.1
• proof
• Example 3.2
• Example 3.3
• Theorem 3.4: Hock:2023qii
• proof
• Remark 3.5