The distribution of the root degree of a random permutation

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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    E. A. Bender: Asymptotical methods in enumeration, Siam Rev. 16 (1974), 485–515.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    J. Blum: Enumeration of the square permutations in S n, J. Comb. Theory (A) 17 (1974), 156–161.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    E. D. Bolker and A. M. Gleason: Counting permutations, J. Comb. Theory (A) 29 (1980), 236–242.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    B. Bollobás: Random Graphs, 2nd Edition, Cambridge Univ. Press (2001).

  5. [5]

    M. Bóna, A. McLennan and D. White: Permutations with roots, Random Structures and Algorithms 17 (2000), 157–167.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    G. H. Hardy and E. M. Wright: An Introduction to the Theory of Numbers, 5th ed., Oxford (1979).

  7. [7]

    N. Pouyanne: On the number of permutations admitting an mth root, Electronic J. Comb. 9 (2002), #R3.

    MathSciNet  Google Scholar 

  8. [8]

    L. A. Shepp and S. P. Lloyd: Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966), 340–357.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    G. Tenenbaum: Introduction to Analysis and Probabilistic Number Theory, Cambridge University Press (1995).

  10. [10]

    P. Turán: On some connections between combinatorics and group theory, Colloq. Math. Soc. János Bolyai (P. Erdős, A. Rényi and V. T. Sós, eds.), pp. 1055–1082, North Holland, Amsterdam (1970).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

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.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Mathematics Subject Classification (2000)

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