Permutations with few inversions are locally uniform

Bevan, David (2019) Permutations with few inversions are locally uniform. Discrete Analysis. ISSN 2397-3129

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    Abstract

    We prove that permutations with few inversions exhibit a local-global 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 k-point 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.