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Generalized Riffle Shuffles and Quasisymmetric Functions
Richard P.Stanley
Department of Mathematics 2-375, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
rstan@math.mit.edu
Annals of Combinatorics 5 (3) p.479-491 September, 2001
AMS Subject Classification: 05B05, 05A05, 60C05
Abstract:
Given a probability distribution on a totally ordered set, we define for each n>1 a related distribution on the symmetric group , called the QS-distribution. It is a generalization of the q-shuffle distribution considered by Bayer, Diaconis, and Fulman. The QS-distribution is closely related to the theory of quasisymmetric functions and symmetric functions. We obtain explicit formulas in terms of quasisymmetric and symmetric functions for the probability that a random permutation from the QS-distribution satisfies various properties, such as having a given descent set, cycle structure, or shape under the Robinson-Schensted-Knuth algorithm.
Keywords: QS-distribution, q-shuffle, quasisymmetric function, major index, RSK algorithm, increasing subsequence

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