(Joint work with João Araújo.) Martin and Sagan defined a permutation group of degree n to be λ-transitive, where λ is a partition of n, if it acts transitively on ordered set-partitions of shape λ. There is an obvious weakening to transitivity on unordered partitions, which by analogy with classical concepts we call λ-homogeneity. This arises in a problem on transformation semigroups. I will prove a partition analogue of the Livingstone–Wagner Theorem by finding all the groups which are λ-homogeneous but not λ-transitive, and explain how this is applied in semigroup theory.