  We study classes $P_{g,T}(\alpha;\beta)$ on the moduli space of stable, genus
g curves with rational tails defined by pushing forward the virtual fundamental
classes of spaces of relative stable maps to an unparameterized projective
line. A comparison with classes $Q_{g,T}$ arising from sections of the
universal Jacobian shows the classes $P_{g,T}(\alpha;\beta)$ are polynomial in
the parts of the partitions indexing the special ramification data. Virtual
localization on moduli spaces of relative stable maps gives sufficient
relations to compute the coefficients of these polynomials in various cases.
