On Promotion and Quasi-Tangled Labelings of Posets

IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED
Eliot Hodges
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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.

Abstract Image

Abstract Image

论姿势的推广与拟纠缠标注
2022 年,迪凡特和克拉维茨引入了扩展推广(表示为 \( \partial \)),这是一种作用于正集标签集的映射。扩展推广是舒岑伯格推广算子的广义化,舒岑伯格推广算子是一个研究得很透彻的映射,它可以对正集的线性扩展集进行置换。众所周知,如果 L 是一个 n 元素正集 P 的标签,那么 \( \partial ^{n-1}(L) \) 就是一个线性扩展。这使得我们可以把 \( \partial \) 看作是 P 的所有标注集合上的一个排序算子,我们把 P 的线性扩展看作是已经排序过的标注。需要对 \( n-1 \) 的应用进行排序的标注称为纠缠标注;需要对 \( n-2 \) 的应用进行排序的标注称为准纠缠标注。我们统计了一类相对较大的poset的准纠缠标签,这一类poset被称为带瘪叶的膨胀根树。给定一个具有唯一最小元素的 n 元素集合,该最小元素具有一个父元素,那么根据上述枚举,这个集合具有 \( 2(n-1)!-(n-2)!\)个准纠缠标签。使用类似的方法,我们概述了一种算法方法来枚举需要对任意固定的( k\in \{1,\ldots ,n-2\} \)应用进行排序的( n-k-1 \)标签。我们还在证明德凡特(Defant)和克拉维茨(Kravitz)关于一个 n 元素正集的最大可能纠缠标签数的猜想方面取得了部分进展。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Annals of Combinatorics
Annals of Combinatorics 数学-应用数学
CiteScore
1.00
自引率
0.00%
发文量
56
审稿时长
>12 weeks
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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