On Non-Squashing Partitions
N. J. A. Sloane, James A. Sellers
Abstract
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.
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