Links and the Diaconis–Graham Inequality

IF 1 2区 数学 Q1 MATHEMATICS Combinatorica Pub Date : 2024-06-27 DOI:10.1007/s00493-024-00107-1
Christopher Cornwell, Nathan McNew
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Abstract

In 1977 Diaconis and Graham proved two inequalities relating different measures of disarray in permutations, and asked for a characterization of those permutations for which equality holds in one of these inequalities. Such a characterization was first given in 2013. Recently, another characterization was given by Woo, using a topological link in \({\mathbb {R}}^3\) that can be associated to the cycle diagram of a permutation. We show that Woo’s characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham’s inequality is directly related to the Euler characteristic of the associated link. This connection provides a new proof of the original result of Diaconis and Graham. We also characterize permutations with a fixed discrepancy in terms of their associated links and find that the stabilized-interval-free permutations are precisely those whose associated links are nonsplit.

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链接与迪亚科尼斯-格雷厄姆不平等现象
1977 年,Diaconis 和 Graham 证明了两个与排列混乱度量有关的不等式,并要求对在其中一个不等式中相等的排列进行描述。他们于 2013 年首次给出了这样的表征。最近,Woo 又给出了另一个表征,使用的是\({\mathbb {R}}^3\) 中的拓扑链接,它可以与一个排列的循环图相关联。我们的研究表明,Woo 的描述还可以进一步扩展:对于任何置换,Diaconis 和 Graham 不等式中的差异都与相关链接的欧拉特征直接相关。这种联系为 Diaconis 和 Graham 的原始结果提供了新的证明。我们还从相关链接的角度描述了具有固定差异的排列组合,并发现无稳定间隔排列组合正是那些相关链接不分裂的排列组合。
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来源期刊
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.
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