Sami Assaf ; Persi Diaconis ; K. Soundararajan
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Riffle shuffles of a deck with repeated cards
dmtcs:2733 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
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https://doi.org/10.46298/dmtcs.2733
Riffle shuffles of a deck with repeated cards
Authors: Sami Assaf 1; Persi Diaconis 2; K. Soundararajan 3
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Sami Assaf;Persi Diaconis;K. Soundararajan
1 Department of Mathematics [MIT]
2 Department of Statistics [Stanford]
3 Department of Mathematics [Stanford]
We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate.
