On non-squashing partitions

A partition n=p1+p2+⋯+pk with 1⩽p1⩽p2⩽⋯⩽pk is called non-squashing if p1+⋯+pj⩽pj+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.