  We give a graph-sum algorithm that expresses any genus-$g$ Gromov-Witten
invariant of the symmetric product orbifold
$\mathrm{Sym}^d\mathbb{P}^r:=[(\mathbb{P}^r)^d/S_d]$ in terms of "Hurwitz-Hodge
integrals" -- integrals over (compactified) Hurwitz spaces. We apply the
algorithm to prove a partial mirror theorem for $\mathrm{Sym}^d\mathbb{P}^r$ in
genus zero. The theorem states that a generating function of Gromov-Witten
invariants of $\mathrm{Sym}^d\mathbb{P}^r$ is equal to an explicit power series
$I_{\mathrm{Sym}^d\mathbb{P}^r},$ conditional upon a conjectural combinatorial
identity. This is a first step towards proving Ruan's Crepant Resolution
Conjecture for the resolution $\mathrm{Hilb}^{(d)}(\mathbb{P}^2)$ of the coarse
moduli space of $\mathrm{Sym}^d\mathbb{P}^2.$
