W-Operator and Differential Equation for 3-Hurwitz Number

TL;DR

This paper introduces a new type of Hurwitz number involving 3-cycles, and uses W-operators to derive a differential equation for its generating function, extending previous work on Hurwitz numbers.

Contribution

The paper develops a differential equation for the generating function of 3-Hurwitz numbers using W-operators, expanding the mathematical framework for these enumerations.

Findings

Derived a differential equation for 3-Hurwitz number generating function

Extended W-operator techniques to new Hurwitz number types

Connected results to previous work on simple Hurwitz numbers

Abstract

We consider a new type of Hurwitz number, the number of ordered transitive factorizations of an arbitrary permutation into d-cycles. In this paper, we focus on the special case d = 3. The minimal number of transitive factorizations of any permutation into 3-cycles has been worked out by David, Goulden and Jackson. Also, such factorizations for transpositions, the case d = 2, have been considered by Crescimanno and Taylor. Goulden and Jackson have proved the differential equation for the generating series of simple Hurwitz numbers. Based on their results, we use W-operator to prove a differential equation for the generating function of the new type Hurwitz number.