On Sloane's generalization of non-squashing stacks of boxes
The authors give the answer to the following box stacking problem: We are given \(n\) boxes of which the \(i\)th one weighs \((m-1)i\) grams (where \(m\geq2\) is a fixed integer) and can support a total weight of \(i\) grams. What is the number \(a_m(n)\) of different ways to build a stack of boxes in which no box will be squashed by the weight of the boxes above it? The case \(m=2\) was studied in detail by \textit{N. J. A. Sloane} and the second author in [Discrete Math. 294, No.~3, 259--274 (2005; Zbl 1128.11046)]. The number \(a_m(n)\) can be shown to be equal to the number of certain \(m\)-non-squashing partitions. Via partition analysis it is shown that for \(m\geq 3\), \[ \sum_{n\geq0} a_m(n)q^n=\frac{1}{(1-q)^2\prod_{i\geq 0}(1-q^{(m-1)m^i})}. \] The paper concludes with some considerations about congruence properties of \(a_m(n)\).
scientific article; zbMATH DE number 5134186