Hurwitz numbers, ribbon graphs, and tropicalization

  title={Hurwitz numbers, ribbon graphs, and tropicalization},
  author={P. Johnson},
  journal={arXiv: Algebraic Geometry},
  • P. Johnson
  • Published 6 March 2013
  • Mathematics
  • arXiv: Algebraic Geometry
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… 

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