  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.
