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 + · · · + pk with 1 ≤ p1 ≤ p2 ≤ · · · ≤ pk wherein p1 + · · · + pj ≤ pj+1 for 1 ≤ 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.