  Using degeneration and Schubert calculus, we consider the problem of
computing the number of linear series of given degree $d$ and dimension $r$ on
a general curve of genus $g$ satisfying prescribed incidence conditions at $n$
points. We determine these numbers completely for linear series of arbitrary
dimension when $d$ is sufficiently large, and for all $d$ when either $r=1$ or
$n=r+2$. Our formulas generalize and give new proofs of recent results of
Tevelev and of Cela-Pandharipande-Schmitt.
