{"title":"Diametric problem for permutations with the Ulam metric (optimal anticodes)","authors":"Pat Devlin, 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 denote the set of permutations on n symbols, and for each , 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 , compared to the best known construction of size .
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.