Diametric problem for permutations with the Ulam metric (optimal anticodes)

IF 0.9 2区 数学 Q2 MATHEMATICS
Pat Devlin, Leo Douhovnikoff
{"title":"Diametric problem for permutations with the Ulam metric (optimal anticodes)","authors":"Pat Devlin,&nbsp;Leo Douhovnikoff","doi":"10.1016/j.jcta.2024.106002","DOIUrl":null,"url":null,"abstract":"<div><div>We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the set of permutations on <em>n</em> symbols, and for each <span><math><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, define their Ulam distance as the number of distinct symbols that must be deleted from each until they are equal. We obtain a near-optimal upper bound on the size of the intersection of two balls in this space, and as a corollary, we prove that a set of diameter at most <em>k</em> has size at most <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>+</mo><mi>C</mi><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></msup><mi>n</mi><mo>!</mo><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>!</mo></math></span>, compared to the best known construction of size <span><math><mi>n</mi><mo>!</mo><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>!</mo></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106002"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524001419","RegionNum":2,"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

We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let Sn denote the set of permutations on n symbols, and for each σ,τSn, define their Ulam distance as the number of distinct symbols that must be deleted from each until they are equal. We obtain a near-optimal upper bound on the size of the intersection of two balls in this space, and as a corollary, we prove that a set of diameter at most k has size at most 2k+Ck2/3n!/(nk)!, compared to the best known construction of size n!/(nk)!.
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
2.90
自引率
9.10%
发文量
94
审稿时长
12 months
期刊介绍: The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信