  We consider the general problem of enumerating branched covers of the
projective line from a fixed general curve subject to ramification conditions
at possibly moving points. Our main computations are in genus 1; the theory of
limit linear series allows one to reduce to this case. We first obtain a simple
formula for a weighted count of pencils on a fixed elliptic curve E, where
base-points are allowed. We then deduce, using an inclusion-exclusion
procedure, formulas for the numbers of maps E->P^1 with moving ramification
conditions. A striking consequence is the invariance of these counts under a
certain involution. Our results generalize work of Harris, Logan, Osserman, and
Farkas-Moschetti-Naranjo-Pirola.
