{"title":"Dyck Paths, Binary Words, and Grassmannian Permutations Avoiding an Increasing Pattern","authors":"Krishna Menon, Anurag Singh","doi":"10.1007/s00026-023-00667-x","DOIUrl":null,"url":null,"abstract":"<div><p>A permutation is called <i>Grassmannian</i> if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021. We continue this work by studying Grassmannian permutations that avoid an increasing pattern. In particular, we count the Grassmannian permutations of size <i>m</i> avoiding the identity permutation of size <i>k</i>, thus solving a conjecture made by Weiner. We also refine our counts to special classes such as odd Grassmannian permutations and Grassmannian involutions. We prove most of our results by relating Grassmannian permutations to Dyck paths and binary words.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 3","pages":"871 - 887"},"PeriodicalIF":0.6000,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00667-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A permutation is called Grassmannian if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021. We continue this work by studying Grassmannian permutations that avoid an increasing pattern. In particular, we count the Grassmannian permutations of size m avoiding the identity permutation of size k, thus solving a conjecture made by Weiner. We also refine our counts to special classes such as odd Grassmannian permutations and Grassmannian involutions. We prove most of our results by relating Grassmannian permutations to Dyck paths and binary words.
如果一个排列组合最多只有一个后裔，那么它就被称为格拉斯曼排列组合。2021 年，Gil 和 Tomasko 开始研究此类排列中的模式规避问题。我们将继续这项工作，研究避免递增模式的格拉斯曼排列。特别是，我们统计了大小为 m 的格拉斯曼排列避免了大小为 k 的同一性排列，从而解决了韦纳提出的一个猜想。我们还细化了对奇数格拉斯曼排列和格拉斯曼渐开线等特殊类别的计数。我们通过将格拉斯曼排列与戴克路径和二元词联系起来来证明我们的大部分结果。
期刊介绍：
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches
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