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arXiv · 2310.14071

An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation

Anant Godbole·Hannah Swickheimer
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Abstract

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.

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Anant Godbole, Hannah Swickheimer. 2023-10-21. An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation. https://doi.org/10.46298/dmtcs.12458

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