In this thesis, we study a problem that is in a sense a reversal of the Hurwitz counting problem. The Hurwitz problem asks: for a generic target --- P^1 with a list of n points q_1,...,q_n --- and partitions sigma_1,...,sigma_n of d, how many degree d covers C->P^1 are there with ramification profile sigma_i over q_i? We turn the problem on its head and ask: for a generic source -- an n-pointed curve (C,p_1,...,p_r) of genus g -- and partitions mu,sigma_1,...,sigma_n of d where mu has length r, how many degree d covers C -> P^1 are there with ramification profile mu over 0 corresponding to a fiber p_1,...,p_r and elsewhere ramification profiles sigma_1,...,sigma_n?

While the enumerative invariants we study bear a similarity to generalized Tevelev degrees, they are more difficult to express in closed form in general. Nonetheless, we establish key results: after proving a closed form result in the case where the ramification profiles sigma_1 and sigma_2 are "even" (consisting of 2,...,2), and we go on to establish recursive formulas to compute invariants where each ramification profile is of the form (x,1,...,1). A special case for n fixed asks: given a generic d-pointed genus 1 curve (E,p_1,...,p_d), how many covers (E,p_1,...,p_d)->(P^1,0) are there with k (unspecified) points of E having ramification index n? We establish a closed form answer for n=3 and expect exponential growth for n>3.