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We give a combinatorial expansion of a Schubert homology class in the affineGrassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This isachieved by studying the combinatorics of a new class of partitions calledk-shapes, which interpolates between k-cores and k+1-cores. We define asymmetric function for each k-shape, and show that they expand positively interms of dual k-Schur functions. We obtain an explicit combinatorialdescription of the expansion of an ungraded k-Schur function into k+1-Schurfunctions. As a corollary, we give a formula for the Schur expansion of anungraded k-Schur function.

k-shape poset and branching of k-Schur functions