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The combinatorics of biased riffle shuffles

Jason Fulman
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Abstract

This paper studies biased riffle shuffles, first defined by Diaconis, Fill, and Pitman. These shuffles generalize the well-studied Gilbert-Shannon-Reeds shuffle and convolve nicely. An upper bound is given for the time for these shuffles to converge to the uniform distribution; this matches lower bounds of Lalley. A careful version of a bijection of Gessel leads to a generating function for cycle structure after one of these shuffles and gives new results about descents in random permutations. Results are also obtained about the inversion and descent structure of a permutation after one of these shuffles.

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Jason Fulman. 1997-12-09. The combinatorics of biased riffle shuffles. https://arxiv.org/abs/math/9712240

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