A Formula about W-operator and Its Application to Hurwitz Number

TL;DR

This paper explores the W-operator's relation to symmetric group elements and provides a new proof for the differential equation governing the generating function of d-Hurwitz numbers, advancing understanding in algebraic combinatorics.

Contribution

It establishes a novel connection between the W-operator and central elements in symmetric groups, offering an alternative proof for the Hurwitz number generating function's differential equation.

Findings

Derived a relation between W-operator W([d]) and symmetric group central elements.

Provided a new proof for the differential equation of the d-Hurwitz generating function.

Enhanced understanding of W-operator applications in Hurwitz number theory.

Abstract

W-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the W-operator to prove a formula for the generating function of the generalized Hurwitz numbers. A special example of the W-operator is the cut-and-join operator. Goulden and Jackson use the cut-and-join operator to calculate the simple Hurwitz number. In this paper, we study the relation between W-operator W([d]) and the central elements in Sn. Based on the relation we find, we give another proof about a differential equation of the generating function of d-Hurwitz number.