{"title":"On d-Permutations and Pattern Avoidance Classes","authors":"Nathan Sun","doi":"10.1007/s00026-024-00695-1","DOIUrl":null,"url":null,"abstract":"<div><p>Multidimensional permutations, or <i>d</i>-permutations, are represented by their diagrams on <span>\\([n]^d\\)</span> such that there exists exactly one point per hyperplane <span>\\(x_i\\)</span> that satisfies <span>\\(x_i= j\\)</span> for <span>\\(i \\in [d]\\)</span> and <span>\\(j \\in [n]\\)</span>. Bonichon and Morel previously enumerated 3-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate 3-permutations avoiding any two fixed patterns of size 3. We further provide a enumerative result relating 3-permutation avoidance classes with their respective recurrence relations. In particular, we show a recurrence relation for 3-permutations avoiding the patterns 132 and 213, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate 3-permutations avoiding three patterns of size 3.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-024-00695-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Multidimensional permutations, or d-permutations, are represented by their diagrams on \([n]^d\) such that there exists exactly one point per hyperplane \(x_i\) that satisfies \(x_i= j\) for \(i \in [d]\) and \(j \in [n]\). Bonichon and Morel previously enumerated 3-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate 3-permutations avoiding any two fixed patterns of size 3. We further provide a enumerative result relating 3-permutation avoidance classes with their respective recurrence relations. In particular, we show a recurrence relation for 3-permutations avoiding the patterns 132 and 213, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate 3-permutations avoiding three patterns of size 3.