On Sloane's generalization of non-squashing stacks of boxes
type: Item
instance of: scholarly article
title: On Sloane's generalization of non-squashing stacks of boxes
zbMATH Open document ID: 1116.05005
author: George E. Andrews
author: James A. Sellers
published in: Discrete Mathematics
publication date: 2007-03-15T00:00:00Z
review text: 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 \(a_m(n)\).
reviewed by: Astrid Reifegerste
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 5134186
zbMATH Keywords: partitions
zbMATH Keywords: non-squashing partitions
zbMATH Keywords: \(m\)-ary partitions
zbMATH Keywords: stacking boxes
zbMATH Keywords: partition analysis
describes a project that uses: Omega
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: j.disc.2006.07.033
OpenAlex ID: W2109307047
cites work: MacMahon's partition analysis: The Omega package
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cites work: Q4820542
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cites work: On non-squashing partitions
DOI: 10.1016/J.DISC.2006.07.033
description: scientific article; zbMATH DE number 5134186