Factorizations of cycles and multi-noded rooted trees

TL;DR

This paper establishes a combinatorial formula for factorizations of cycles into prescribed cycle lengths, introduces multi-noded rooted trees to model these factorizations, and connects the results to Hurwitz numbers in geometry.

Contribution

It introduces multi-noded rooted trees and proves a bijection with cycle factorizations, providing a new combinatorial approach to a class of Hurwitz numbers.

Findings

Number of factorizations is d^{r-2} under certain conditions

Introduces multi-noded rooted trees as a new combinatorial object

Connects cycle factorizations to Hurwitz numbers in algebraic geometry

Abstract

In this paper, we study factorizations of cycles. The main result is that under certain condition, the number of ways to factor a $d$-cycle into a product of cycles of prescribed lengths is $d^{r-2}.$ To prove our result, we first define a new class of combinatorial objects, multi-noded rooted trees, which generalize rooted trees. We find the cardinality of this new class which with proper parameters is exactly $d^{r-2}.$ The main part of this paper is the proof that there is a bijection from factorizations of a $d$-cycle to multi-noded rooted trees via factorization graphs. This implies the desired formula. The factorization problem we consider has its origin in geometry, and is related to the study of a special family of Hurwitz numbers: pure-cycle Hurwitz numbers. Via the standard translation of Hurwitz numbers into group theory, our main result is equivalent to the following: when the genus is $0$ and one of the ramification indices is $d,$ the degree of the covers, the pure-cycle Hurwitz number is $d^{r-3},$ where $r$ is the number of branch points.