{"title":"On Promotion and Quasi-Tangled Labelings of Posets","authors":"Eliot Hodges","doi":"10.1007/s00026-023-00646-2","DOIUrl":null,"url":null,"abstract":"<div><p>In 2022, Defant and Kravitz introduced extended promotion (denoted <span>\\( \\partial \\)</span>), a map that acts on the set of labelings of a poset. Extended promotion is a generalization of Schützenberger’s promotion operator, a well-studied map that permutes the set of linear extensions of a poset. It is known that if <i>L</i> is a labeling of an <i>n</i>-element poset <i>P</i>, then <span>\\( \\partial ^{n-1}(L) \\)</span> is a linear extension. This allows us to regard <span>\\( \\partial \\)</span> as a sorting operator on the set of all labelings of <i>P</i>, where we think of the linear extensions of <i>P</i> as the labelings which have been sorted. The labelings requiring <span>\\( n-1 \\)</span> applications of <span>\\( \\partial \\)</span> to be sorted are called <i>tangled</i>; the labelings requiring <span>\\( n-2 \\)</span> applications are called <i>quasi-tangled</i>. We count the quasi-tangled labelings of a relatively large class of posets called <i>inflated rooted trees with deflated leaves</i>. Given an <i>n</i>-element poset with a unique minimal element with the property that the minimal element has exactly one parent, it follows from the aforementioned enumeration that this poset has <span>\\( 2(n-1)!-(n-2)! \\)</span> quasi-tangled labelings. Using similar methods, we outline an algorithmic approach to enumerating the labelings requiring <span>\\( n-k-1 \\)</span> applications to be sorted for any fixed <span>\\( k\\in \\{1,\\ldots ,n-2\\} \\)</span>. We also make partial progress towards proving a conjecture of Defant and Kravitz on the maximum possible number of tangled labelings of an <i>n</i>-element poset.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 2","pages":"529 - 554"},"PeriodicalIF":0.6000,"publicationDate":"2023-04-20","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/s00026-023-00646-2","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
In 2022, Defant and Kravitz introduced extended promotion (denoted \( \partial \)), a map that acts on the set of labelings of a poset. Extended promotion is a generalization of Schützenberger’s promotion operator, a well-studied map that permutes the set of linear extensions of a poset. It is known that if L is a labeling of an n-element poset P, then \( \partial ^{n-1}(L) \) is a linear extension. This allows us to regard \( \partial \) as a sorting operator on the set of all labelings of P, where we think of the linear extensions of P as the labelings which have been sorted. The labelings requiring \( n-1 \) applications of \( \partial \) to be sorted are called tangled; the labelings requiring \( n-2 \) applications are called quasi-tangled. We count the quasi-tangled labelings of a relatively large class of posets called inflated rooted trees with deflated leaves. Given an n-element poset with a unique minimal element with the property that the minimal element has exactly one parent, it follows from the aforementioned enumeration that this poset has \( 2(n-1)!-(n-2)! \) quasi-tangled labelings. Using similar methods, we outline an algorithmic approach to enumerating the labelings requiring \( n-k-1 \) applications to be sorted for any fixed \( k\in \{1,\ldots ,n-2\} \). We also make partial progress towards proving a conjecture of Defant and Kravitz on the maximum possible number of tangled labelings of an n-element poset.
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