  We give a combinatorial expansion of a Schubert homology class in the affine
Grassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This is
achieved by studying the combinatorics of a new class of partitions called
k-shapes, which interpolates between k-cores and k+1-cores. We define a
symmetric function for each k-shape, and show that they expand positively in
terms of dual k-Schur functions. We obtain an explicit combinatorial
description of the expansion of an ungraded k-Schur function into k+1-Schur
functions. As a corollary, we give a formula for the Schur expansion of an
ungraded k-Schur function.
