  Weighted constellations give graphical representations of weighted branched
coverings of the Riemann sphere. They were introduced to provide a
combinatorial interpretation of the $2$D Toda $\tau$-functions of
hypergeometric type serving as generating functions for weighted Hurwitz
numbers in the case of polynomial weight generating functions. The product over
all vertex and edge weights of a given weighted constellation, summed over all
configurations, reproduces the $\tau$-function. In the present work, this is
generalized to constellations in which the weighting parameters are determined
by a rational weight generating function. The associated $\tau$-function may be
expressed as a sum over the weights of doubly labelled weighted constellations,
with two types of weighting parameters associated to each equivalence class of
branched coverings. The double labelling of branch points, referred to as
"colour" and "flavour" indices, is required by the fact that, in the Taylor
expansion of the weight generating function, a particular colour from amongst
the denominator parameters may appear multiply, and the flavour labels indicate
this multiplicity.
