{"title":"Avoiding monotone arithmetic progressions in permutations of integers","authors":"Sarosh Adenwalla","doi":"10.1016/j.disc.2024.114183","DOIUrl":null,"url":null,"abstract":"<div><p>A permutation of the integers avoiding monotone arithmetic progressions of length 6 was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length 5. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length 4. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length 5. A permutation of the positive integers that avoided monotone arithmetic progressions of length 4 with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length 4 with common difference not divisible by <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. In addition, we specify the structure of permutations of <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span> that avoid length 3 monotone arithmetic progressions mod <em>n</em> as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length <em>k</em> monotone arithmetic progressions mod <em>n</em>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"347 11","pages":"Article 114183"},"PeriodicalIF":0.7000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003145/pdfft?md5=8d8c7e64880fe3b6fb3a3eb024d87635&pid=1-s2.0-S0012365X24003145-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003145","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A permutation of the integers avoiding monotone arithmetic progressions of length 6 was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length 5. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length 4. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length 5. A permutation of the positive integers that avoided monotone arithmetic progressions of length 4 with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each , there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length 4 with common difference not divisible by . In addition, we specify the structure of permutations of that avoid length 3 monotone arithmetic progressions mod n as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length k monotone arithmetic progressions mod n.
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.