W-Operator and Differential Equation for 3-Hurwitz Number
Combinatorics
2016-12-12 v1
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.
Keywords
Cite
@article{arxiv.1612.02884,
title = {W-Operator and Differential Equation for 3-Hurwitz Number},
author = {Hao Sun},
journal= {arXiv preprint arXiv:1612.02884},
year = {2016}
}