Block-transitive designs with a poset of imprimitive partitions

We study block designs which admit an automorphism group that is transitive on blocks and points, and leaves invariant every partition in a given finite poset of partitions of the point set. The full stabiliser $G$ of all the partitions in the poset is a generalised wreath product. We use the theory of generalised wreath products to give necessary and sufficient conditions, in terms of the `array' of a point-subset $B$, for the set of $G$-images of $B$ to form the block-set of a $G$-block-transitive $2$-design. This generalises previous results for the special cases where the poset is a chain or an anti-chain. We also give explicit infinite families of examples of $2$-designs for each poset involving three proper partitions, and for the famous $N$-poset with four partitions. (Posets with two proper partitions have been treated previously.) This suggests the problem of finding explicit examples for other posets.