W-Operator and Hurwitz Number

Hao Sun

Published 2016-11-15Version 1

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 W-operators 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 CS_n. We also give another proof for the formula proved by Mironov et al..