The Expected Number of Distinct Consecutive Patterns in a Random Permutation

Austin Allen; Dylan Cruz Fonseca; Veronica Dobbs; Egypt Downs; Evelyn Fokuoh; Anant Godbole; Sebastián Papanikolaou Costa; Christopher Soto; Lino Yoshikawa

The Expected Number of Distinct Consecutive Patterns in a Random Permutation

Combinatorics 2020-11-25 v1 Probability

Authors: Austin Allen , Dylan Cruz Fonseca , Veronica Dobbs , Egypt Downs , Evelyn Fokuoh , Anant Godbole , Sebastián Papanikolaou Costa , Christopher Soto , Lino Yoshikawa

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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, we show that the expected number of distinct consecutive patterns in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ . This exhibits the fact that random permutations pack consecutive patterns near-perfectly.

Keywords

permutations pattern avoidance in permutations combinatorial bounds and extremal problems

Cite

@article{arxiv.2011.12179,
  title  = {The Expected Number of Distinct Consecutive Patterns in a Random Permutation},
  author = {Austin Allen and Dylan Cruz Fonseca and Veronica Dobbs and Egypt Downs and Evelyn Fokuoh and Anant Godbole and Sebastián Papanikolaou Costa and Christopher Soto and Lino Yoshikawa},
  journal= {arXiv preprint arXiv:2011.12179},
  year   = {2020}
}

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12 pages, 2 figures

The Expected Number of Distinct Consecutive Patterns in a Random Permutation

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