On Sloaneʹs 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 image with image wherein image for image. 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.