On non-squashing partitions

N. J.A. Sloane; James A. Sellers

doi:10.1016/j.disc.2004.11.014

On non-squashing partitions

N. J.A. Sloane, James A. Sellers

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

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.

Original language	English (US)
Pages (from-to)	259-274
Number of pages	16
Journal	Discrete Mathematics
Volume	294
Issue number	3
DOIs	https://doi.org/10.1016/j.disc.2004.11.014
State	Published - May 6 2005
Externally published	Yes

Keywords

Binary partitions
Non-squashing partitions
Partitions
Stacking boxes
m-ary Partitions

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10.1016/j.disc.2004.11.014

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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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AU - Sloane, N. J.A.

AU - Sellers, James A.

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N2 - 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.

AB - 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.

KW - Binary partitions

KW - Non-squashing partitions

KW - Partitions

KW - Stacking boxes

KW - m-ary Partitions

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DO - 10.1016/j.disc.2004.11.014

M3 - Article

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SN - 0012-365X

VL - 294

SP - 259

EP - 274

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 3

ER -

On non-squashing partitions

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