  We study some aspects of the $\lambda_g$ pairing on the tautological ring of
$M_g^c$, the moduli space of genus $g$ stable curves of compact type. We
consider pairing kappa classes with pure boundary strata, all tautological
classes supported on the boundary, or the full tautological ring. We prove that
the rank of this restricted pairing is equal in the first two cases and has an
explicit formula in terms of partitions, while in the last case the rank
increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on
the tautological ring of $M_g$.
