By an extension of steepest-descent methods for "Riemann-Hilbert" problems specifically designed to handle analytic functions without jumps along contours but instead having a large number of prescribed singularities, we will show how to compute strong large-degree asymptotics for polynomials orthogonal with respect to measures made up solely of a large number of point masses.  Among other applications, the resulting asymptotic formulae can be used to compute statistics associated with certain measures on random partitions.  This is joint work with Jinho Baik, Thomas Kriecherbauer, and Ken McLaughlin.