Riffle shuffles, cycles, and descents

Abstract

We derive closed form expressions and limiting formulae for a variety of functions of a permutation resulting from repeated riffle shuffles. The results allow new formulae and approximations for the number of permutations inS n with given cycle type and number of descents. The theorems are derived from a bijection discovered by Gessel. A self-contained proof of Gessel's result is given.

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Diaconis, P., Grath, M.M. & Pitman, J. Riffle shuffles, cycles, and descents. Combinatorica 15, 11–29 (1995). https://doi.org/10.1007/BF01294457

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Mathematics Subject Classification (1991)

  • 05 A 15
  • 60 C 05