Bi-pruned Hurwitz numbers

TL;DR

This paper introduces bi-pruned double Hurwitz numbers, a new subset of ramified coverings that simplifies the enumeration while retaining key properties, providing insights into the structure of double Hurwitz numbers.

Contribution

It defines bi-pruned double Hurwitz numbers, establishing their properties and their role as a core subset that determines the full double Hurwitz numbers.

Findings

Bi-pruned numbers are smaller but determine double Hurwitz numbers.

They exhibit piecewise polynomial behavior.

They can be expressed in the symmetric group.

Abstract

Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary ramification over $0$ and $\infty$ and simple ramification over $b$ points, where $b$ is given by the Riemann-Hurwitz formula. In this work, we introduce the notion of bi-pruned double Hurwitz numbers. This is a new enumerative problem, which yields smaller numbers but completely determines double Hurwitz numbers. They count a relevant subset of covers and share many properties with double Hurwitz numbers, such as piecewise polynomial behaviour and an expression in the symmetric group. Thus, we may view them as a core of the double Hurwitz numbers problem. This work is built on and generalises previous work of Do--Norbury and the author.