Integrality of the simple Hurwitz numbers

TL;DR

This paper proves that simple Hurwitz numbers are always integers using a combinatorial approach centered on the cut-and-join operator.

Contribution

It provides a new combinatorial proof of the integrality of simple Hurwitz numbers, enhancing understanding of their mathematical structure.

Findings

Confirmed integrality of simple Hurwitz numbers

Developed combinatorial proof method

Utilized cut-and-join operator in proof

Abstract

We show the integrality of the simple Hurwitz numbers. The main tool is the cut-and-join operator, and our proof is a purely combinatorial one.