The Combinatorics of Real Double Hurwitz Numbers with Real Positive Branch Points

We investigate the combinatorics of real double Hurwitz numbers with real positive branch points using the symmetric group. Our main focus is 2-fold. First, we prove correspondence theorems relating these numbers to counts of tropical real covers and study the structure of real double Hurwitz numbers with the help of the tropical count. Second, we express the numbers as counts of paths in a subgraph of the Cayley graph of the symmetric group. By restricting to real double Hurwitz numbers with real positive branch points, we obtain a concise translation of the counting problem in terms of tuples of elements of the symmetric group that enables us to uncover the beautiful combinatorics of these numbers both in tropical geometry and in the Cayley graph.