# Hurwitz numbers, ribbon graphs, and tropicalization

@article{Johnson2012HurwitzNR, title={Hurwitz numbers, ribbon graphs, and tropicalization}, author={P. Johnson}, journal={arXiv: Algebraic Geometry}, year={2012} }

The double Hurwitz number Hg(µ, ν) has at least four equivalent definitions.
Most naturally, it counts the covers of the Riemann sphere by genus g
curves with certain specified ramification data. This is classically equivalent to
counting certain collections of permutations. More recently, it has been shown to
be equivalent to a count of certain ribbon graphs, or as a weighted count of certain
labeled graphs.
This note is an expository account of the equivalences between these definitions…

## 8 Citations

Combinatorics of tropical Hurwitz cycles

- Mathematics
- 2014

We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple…

Bijections for simple and double Hurwitz numbers

- Mathematics
- 2014

We give a bijective proof of Hurwitz formula for the number of simple branched coverings of the sphere by itself. Our approach extends to double Hurwitz numbers and yields new properties for them. In…

Tropical covers of curves and their moduli spaces

- Mathematics
- 2015

We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications.…

Hurwitz theory and the double ramification cycle

- Mathematics
- 2014

This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov–Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and…

Pruned Double Hurwitz Numbers

- MathematicsElectron. J. Comb.
- 2017

It can be said that pruned double Hurwitz numbers count a relevant subset of covers, leading to considerably smaller numbers and computations, but still featuring the important properties the authors can observe for doubleHurwitz numbers.

Algorithmic aspects of tropical intersection theory

- Mathematics
- 2014

In the first part of this thesis we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using…

Dessins, their delta-matroids and partial duals

- MathematicsSIGMAP 2016
- 2014

It is proved that every map has a partial dual defined over its field of moduli.

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