  Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed
ramification data. Certain kinds of ramification data are of particular
interest, such as double Hurwitz numbers, which count covers with fixed
arbitrary ramification over $0$ and $\infty$ and simple ramification over $b$
points, where $b$ is given by the Riemann-Hurwitz formula. In this work, we
introduce the notion of bi-pruned double Hurwitz numbers. This is a new
enumerative problem, which yields smaller numbers but completely determines
double Hurwitz numbers. They count a relevant subset of covers and share many
properties with double Hurwitz numbers, such as piecewise polynomial behaviour
and an expression in the symmetric group. Thus, we may view them as a core of
the double Hurwitz numbers problem. This work is built on and generalises
previous work of Do--Norbury and the author.
