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dc.contributor.authorDalfó, Cristina
dc.contributor.authorFiol, Miguel Angel
dc.contributor.authorLópez Lorenzo, Ignacio
dc.date.accessioned2021-03-16T11:24:54Z
dc.date.available2021-03-16T11:24:54Z
dc.date.issued2021
dc.identifier.issn0166-218X
dc.identifier.urihttp://hdl.handle.net/10459.1/70756
dc.description.abstractThe Mondrian problem consists of dissecting a square of side length n ∈ N into non-congruent rectangles with natural length sides such that the difference d(n) between the largest and the smallest areas of the rectangles partitioning the square is minimum. In this paper, we compute some bounds on d(n) in terms of the number of rectangles of the square partition. These bounds provide us optimal partitions for some values of n ∈ N. We provide a sequence of square partitions such that d(n)/n2 tends to zero for n large enough. For the case of 'perfect' partitions, that is, with d(n) = 0, we show that, for any fixed powers s1, . . . , sm, a square with side length n = p s1 1 · · · p smm , can have a perfect Mondrian partition only if p1 satisfies a given lower bound. Moreover, if n(x) is the number of side lengths x (with n ≤ x) of squares not having a perfect partition, we prove that its 'density' n(x) x is asymptotic to (log(log(x))2 2 log x , which improves previous results.
dc.description.sponsorshipThe research of C. Dalfó and N. López has been partially supported by grant MTM2017-86767-R (Spanish Ministerio de Ciencia e Innovación). The research of C. Dalfó and M. A. Fiol has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by Spanish Ministerio de Ciencia e Innovación from the Spanish Government under project PGC2018-095471-B-I00. The research of the first author has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922.
dc.format.mimetypeapplication/pdf
dc.language.isoeng
dc.publisherElsevier
dc.relationMINECO/PN2013-2016/MTM2017-86767-R
dc.relationMINECO/PN2017-2020/PGC2018-095471-B-I00
dc.relation.isformatofVersió preprint del document publicat a: https://doi.org/10.1016/j.dam.2021.01.016
dc.relation.ispartofDiscrete Applied Mathematics, 2021, vol. 293, p. 64-73
dc.rightscc-by-nc-nd (c) Elsevier, 2021
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subjectNon-congruent rectangles
dc.subjectPartition
dc.subjectMondrian problem
dc.titleNew results for the Mondrian art problem
dc.typeinfo:eu-repo/semantics/article
dc.date.updated2021-03-16T11:24:54Z
dc.identifier.idgrec030976
dc.type.versioninfo:eu-repo/semantics/acceptedVersion
dc.rights.accessRightsinfo:eu-repo/semantics/openAccess
dc.identifier.doihttps://doi.org/10.1016/j.dam.2021.01.016
dc.relation.projectIDinfo:eu-repo/grantAgreement/EC/H2020/734922/EU/CONNECT
dc.date.embargoEndDate2023-01-30


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cc-by-nc-nd (c) Elsevier, 2021
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