{"title":"Latin hypercubes realizing integer partitions","authors":"Diane Donovan, Tara Kemp, James Lefevre","doi":"10.1016/j.disc.2024.114333","DOIUrl":null,"url":null,"abstract":"<div><div>For an integer partition <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>N</mi></math></span>, a 2-realization of this partition is a latin square of order <em>N</em> with disjoint subsquares of orders <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to <em>m</em>-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114333"},"PeriodicalIF":0.7000,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004643","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
For an integer partition , a 2-realization of this partition is a latin square of order N with disjoint subsquares of orders . The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to m-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.
期刊介绍：
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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