On Non-Squashing Partitions

A partition n = p_1 + p_2 + … + p_k with 1 <= p_1 <= p_2 <= … <= p_k is called non-squashing if p_1 + … + p_j <= p_{j+1} for 1 <= j <= k-1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the number of binary partitions of n. Here we exhibit an explicit bijection between the two families, and determine the number of non-squashing partitions with distinct parts, with a specified number of parts, or with a specified maximal part. We use the results to solve a certain box-stacking problem.