The combinatorics of real double Hurwitz numbers with real positive branch points

TL;DR

This paper explores the combinatorial structure of real double Hurwitz numbers with positive branch points, establishing connections with tropical geometry and Cayley graph paths to deepen understanding of their enumeration.

Contribution

It introduces new correspondence theorems linking Hurwitz numbers to tropical covers and Cayley graph paths, revealing their combinatorial nature in these frameworks.

Findings

Established correspondence between Hurwitz numbers and tropical real covers

Expressed Hurwitz numbers as counts of paths in Cayley graph subgraphs

Uncovered combinatorial structures underlying real double Hurwitz numbers

Abstract

We investigate the combinatorics of real double Hurwitz numbers with real positive branch points using the symmetric group. Our main focus is twofold. First, we prove correspondence theorems relating these numbers to counts of tropical real covers and study the structure of real double Hurwitz numbers with the help of the tropical count. Second, we express the numbers as counts of paths in a subgraph of the Cayley graph of the symmetric group. By restricting to real double Hurwitz numbers with real positive branch points, we obtain a concise translation of the counting problem in terms of tuples of elements of the symmetric group that enables us to uncover the beautiful combinatorics of these numbers both in tropical geometry and in the Cayley graph.