aequationes mathematicae

, Volume 22, Issue 1, pp 268–292 | Cite as

The bias of three pseudo-random shuffles

  • D. P. Robbins
  • E. D. Bolker
Research Papers


Three schemes for shuffling a deck ofn cards are studied, each involving a random choice from [n] n . The shuffles favor some permutations over others sincen! does not dividen n . The probabilities that the shuffles lead to some simple permutations, for instance cycles left and right and the identity, are calculated. Some inequalities are obtained which lead to information about the least and most likely permutations. Numbers of combinatorial interest occur, notably the Catalan numbers and the Bell numbers.

AMS (1980) subject classification

Primary 05A15 Secondary 05C30, 05A17 


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    Knuth, D.,The art of computer programming. Addison Wesley, Reading, Vol. 2, 1969, Vol. 3, 1973.Google Scholar
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    Lovász, L.,Combinatorial problems and exercises. North-Holland, New York, 1979.Google Scholar

Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • D. P. Robbins
    • 1
    • 2
  • E. D. Bolker
    • 1
    • 2
  1. 1.Department of MathematicsWashington and Lee UniversityLexingtonUSA
  2. 2.Department of MathematicsUniversity of MassachusettsBostonUSA

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