On Sloane's generalization of non-squashing stacks of boxes

Recently, Sloane and Sellers solved a certain box stacking problem related to non-squashing   partitions. These are defined as partitions n=p1+p2+⋯+pkn=p1+p2+⋯+pk with 1⩽p1⩽p2⩽⋯⩽pk1⩽p1⩽p2⩽⋯⩽pk wherein p1+⋯+pj⩽pj+1p1+⋯+pj⩽pj+1 for 1⩽j⩽k-11⩽j⩽k-1. Sloane has also hinted at a generalized box stacking problem which is closely related to generalized non-squashing partitions. We solve this generalized box stacking problem by obtaining a generating function for the number of such stacks and discuss partition functions which arise via this generating function.