The distribution of the root degree of a random permutation


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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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).

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Mathematics Subject Classification (2000)

  • 05A15
  • 05A16
  • 60C05
  • 60F05