Abacus models for parabolic quotients of affine Weyl groups

We introduce abacus diagrams that describe the minimal length coset representatives of affine Weyl groups in types , , and . These abacus diagrams use a realization of the affine Weyl group due to Eriksson to generalize a construction of James for the symmetric group. We also describe several combinatorial models for these parabolic quotients that generalize classical results in type related to core partitions.