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The distribution of the root degree of a random permutation

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Abstract

Given a permutation ω of {1, …, n}, let R(ω) be the root degree of ω, i.e. the smallest (prime) integer r such that there is a permutation σ with ω = σ r. We show that, for ω chosen uniformly at random, R(ω) = (lnlnn − 3lnlnln n + O p(1))−1 lnn, and find the limiting distribution of the remainder term.

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Authors and Affiliations

  1. Department of Pure Mathematics & Mathematical Statistics, University of Cambridge Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

    Béla Bollobás

  2. Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, 38152-3240, USA

    Béla Bollobás

  3. Department of Mathematics, Ohio State University, Columbus, Ohio, 43210-1174, USA

    Boris Pittel

Authors
  1. Béla Bollobás
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  2. Boris Pittel
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Correspondence to Boris Pittel.

Additional information

Research supported in part by NSF grants CCR-0225610, DMS-0505550 and ARO grant W911NF-06-1-0076.

Research supported by NSF grant DMS-0406024.

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Bollobás, B., Pittel, B. The distribution of the root degree of a random permutation. Combinatorica 29, 131–151 (2009). https://doi.org/10.1007/s00493-009-2343-3

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