The Annals of Probability

Cycle structure of riffle shuffles

Steven P. Lalley

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Abstract

A class of models for riffle shuffles ("$f$-shuffles") related to certain expansive mappings of the unit interval is studied. The main result concerns the cycle structure of the resulting random permutations in $\mathscr{S}_n$ when n is large. It describes the asymptotic distribution of the number of cycles of a given length, relating this distribution to dynamical properties of the associated mapping. This result generalizes a recent result of Diaconis, McGrath and Pitman.

Article information

Source
Ann. Probab. Volume 24, Number 1 (1996), 49-73.

Dates
First available in Project Euclid: 15 January 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1042644707

Mathematical Reviews number (MathSciNet)
MR1387626

Digital Object Identifier
doi:10.1214/aop/1042644707

Zentralblatt MATH identifier
0854.05007

Subjects
Primary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 60C05

Keywords
Riffle shuffle random permutation interval mapping

Citation

Lalley, Steven P. Cycle structure of riffle shuffles. Ann. Probab. 24 (1996), no. 1, 49--73. doi:10.1214/aop/1042644707. http://projecteuclid.org/euclid.aop/1042644707.


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References

  • [1] Bay er, D. and Diaconis, P. (1992). Trailing the dovetail shuffle to its lair. Ann. Appl. Probab. 2 294-313.
  • [2] Devaney, R. L. (1989). An Introduction to Chaotic Dy namical Sy stems. Addison-Wesley, Reading, MA.
  • [3] Diaconis, P., McGrath, M. and Pitman, J. (1995). Riffle shuffles, cy cles, and descents. Combinatorica 15 11-29.
  • [4] Gessel, I. and Reutenauer, C. (1993). Counting permutations with given cy cle structure and descent set. J. Combin. Theory Ser. A 64 189-215.
  • [5] Kingman, J. (1977). The population structure associated with the Ewens sampling formula. Theoret. Population Biol. 11 274-283.
  • [6] Lalley, S. (1994). Riffle shuffles and dy namical sy stems on the unit interval. Technical report, Dept. Statistics, Purdue Univ.
  • [7] Lloy d, S. P. and Shepp, L. (1966). Ordered cy cle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340-357.
  • [8] Reeds, J. (1981). Unpublished manuscript.
  • [9] Vershik, A. M. and Schmidt, A. (1977). Limit measures arising in asy mptotic theory of sy mmetric groups. Probability Theory and Applications 22 72-88; 23 34-46.