An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation
| Autor: | Godbole, Anant, Swickheimer, Hannah |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Discrete Mathematics & Theoretical Computer Science, vol. 26:1, Permutation Patterns 2023, Special issues (May 3, 2024) dmtcs:12458 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.46298/dmtcs.12458 |
| Popis: | 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. Comment: 11 pages |
| Databáze: | arXiv |
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