Permutations with few inversions are locally uniform
Bevan, David (2019) Permutations with few inversions are locally uniform. Discrete Analysis. ISSN 23973129
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Abstract
We prove that permutations with few inversions exhibit a localglobal dichotomy in the following sense. Suppose σ is a permutation chosen uniformly at random from the set of all permutations of [n] with exactly m = m(n) ≪ n2 inversions. If i < j are chosen uniformly at random from [n], then σ (i) < σ (j) asymptotically almost surely. However, if i and j are chosen so that j  i ≪ m/n, and m ≪ n2/log2n, then limn→∞ P [ σ (i) < σ (j)] = ½. Moreover, if k = k(n) ≪ √(m/n), then the restriction of σ to a random kpoint interval is asymptotically uniformly distributed over Sk. Thus, knowledge of the local structure of σ reveals nothing about its global form. We establish that √(m/n) is the threshold for local uniformity and m/n the threshold for inversions, and determine the behaviour in the critical windows.


Item type: Article ID code: 72724 Dates: DateEvent20 August 2019Published20 August 2019SubmittedSubjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 11 Jun 2020 13:41 Last modified: 01 Aug 2024 01:26 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/72724