Indecomposable decompositions of torsion-free abelian groups

An indecomposable decomposition of a torsion-free abelian group G of rank n is a decomposition G=A₁⊕⋯⊕A_t where each A_i is indecomposable of rank r_i, so that ∑_ir_i=n is a partition of n. The group G may have indecomposable decompositions that result in different partitions of n. We address the problem of characterizing those sets of partitions of n which can arise from indecomposable decompositions of a torsion-free abelian group.