  We extract a system of numerical invariants from logarithmic intersection
theory on pluricanonical double ramification cycles, and show that these
invariants exhibit a number of properties that are enjoyed by double Hurwitz
numbers. Among their properties are (i) the numbers can be efficiently
calculated by counts of tropical curves with a modified balancing condition,
(ii) they are piecewise polynomial in the entries of the ramification vector,
and (iii) they are matrix elements of operators on the Fock space. The numbers
are extracted from the logarithmic double ramification cycle, which is a lift
of the standard double ramification cycle to a blowup of the moduli space of
curves. The blowup is determined by tropical geometry. We show that the
traditional double Hurwitz numbers are intersections of the refined cycle with
the cohomology class of a piecewise polynomial function on the tropical moduli
space of curves. This perspective then admits a natural, combinatorially
motivated, generalization to the pluricanonical setting. Tropical
correspondence results for the new invariants lead immediately to the
structural results for these numbers.
