Packing densities of more 2-block patterns

In this paper, we generalize the inductive techniques used in a previous paper [D. Warren, Optimal packing behavior of some 2-block patterns, Ann. Combin. 8 (2) (2004) 355–367] to prove a general result by which we may calculate the packing density of a 2-block pattern given the packing behavior of a smaller pattern having the same ratio of block sizes. We apply this result to compute the packing densities of patterns having layer sequences of the form (α1,β) in the case that β⩾α⩾2 and α|2β. Necessary unimodality and maximality results are proven as we need them.