  The $P = W$ conjecture identifies the perverse filtration of the Hitchin
system on the cohomology of the moduli space of Higgs bundles with the weight
filtration of the corresponding character variety. In this paper, we introduce
an enumerative approach to to this problem; our technique only uses the
structure of the equivariant intersection numbers on the moduli space of Higgs
bundles, and little information about the topology of the Hitchin map. In the
rank $2$ case, starting from the known intersection numbers of the moduli of
stable bundles, we derive the equivariant intersection numbers on the Higgs
moduli, and then verify the top perversity part of our enumerative $P = W$
statement for even tautological classes. A key in this calculation is the
existence of polynomial solutions to the Discrete Heat Equation satisfying
particular vanishing properties. For odd classes, we derive a determinantal
criterion for the enumerative $P = W$.
