  In this paper we use generating function methods to obtain new asymptotic
results about spaces of $F$-stable maximal tori in $GL_n(\overline{F_q})$,
$Sp_{2n}(\overline{F_q})$, and $SO_{2n+1}(\overline{F_q})$. We recover
stability results of Church--Ellenberg--Farb and Jim\'enez Rolland--Wilson for
"polynomial" statistics on these spaces, and we compute explicit formulas for
their stable values. We derive a double generating function for the characters
of the cohomology of flag varieties in type B/C, which we use to obtain analogs
in type B/C of results of Chen: we recover "twisted homological stability" for
the spaces of maximal tori in $Sp_{2n}(\mathbb{C})$ and
$SO_{2n+1}(\mathbb{C})$, and we compute a generating function for their "stable
twisted Betti numbers". We also give a new proof of a result of Lehrer using
symmetric function theory.
