Elsevier

Discrete Mathematics

Volume 312, Issue 10, 28 May 2012, Pages 1692-1698
Discrete Mathematics

Universal number partition problem with divisibility

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Abstract

We examine a version of the Universal Number Partition Problem with a divisibility property referred to as the Universal Shelf Packing Problem (USPP). We show that if a shelf length is a product of powers of two primes the USPP is always partitionable. In the case where the shelf length is a product of three distinct primes we propose an efficient scheme to determine when such a case is not partitionable. We also show that a shelf length that is a product of powers of four or more primes always has at least one partition failure. Our analysis uses elementary number theory, known results related to the linear Diophantine Frobenius problem, and a new result on Frobenius gaps.

Keywords

Integer partitions
Packing
Frobenius problem

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