Ben Drucker, Eli Garcia, Emily Gunawan, Aubrey Rumbolt, Rose Silver
{"title":"RSK tableaux and box-ball systems","authors":"Ben Drucker, Eli Garcia, Emily Gunawan, Aubrey Rumbolt, Rose Silver","doi":"10.5070/c63261978","DOIUrl":null,"url":null,"abstract":"A box-ball system is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on \\(n\\) objects gives a box-ball system state by assigning its one-line notation to \\(n\\) consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. We prove that if the soliton decomposition of a permutation \\(w\\) is a standard tableau or if its shape coincides with the Robinson-Schensted (RS) partition of \\(w\\), then the soliton decomposition of \\(w\\) and the RS insertion tableau of \\(w\\) are equal. We also use row reading words, Knuth moves, RS recording tableaux, and a localized version of Greene's theorem (proven recently by Lewis, Lyu, Pylyavskyy, and Sen) to study various properties of a box-ball system.Mathematics Subject Classifications: 05A05, 05A17, 37B15Keywords: Permutations, box-ball systems, soliton cellular automata, Young tableaux, Robinson-Schensted-Knuth correspondence, Greene's theorem, Knuth equivalence","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5070/c63261978","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: 1
Abstract
A box-ball system is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on \(n\) objects gives a box-ball system state by assigning its one-line notation to \(n\) consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. We prove that if the soliton decomposition of a permutation \(w\) is a standard tableau or if its shape coincides with the Robinson-Schensted (RS) partition of \(w\), then the soliton decomposition of \(w\) and the RS insertion tableau of \(w\) are equal. We also use row reading words, Knuth moves, RS recording tableaux, and a localized version of Greene's theorem (proven recently by Lewis, Lyu, Pylyavskyy, and Sen) to study various properties of a box-ball system.Mathematics Subject Classifications: 05A05, 05A17, 37B15Keywords: Permutations, box-ball systems, soliton cellular automata, Young tableaux, Robinson-Schensted-Knuth correspondence, Greene's theorem, Knuth equivalence
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.