https://hal.inria.fr/hal-01185425Assaf, SamiSamiAssafDepartment of Mathematics [MIT] - MIT - Massachusetts Institute of TechnologyDiaconis, PersiPersiDiaconisDepartment of Statistics [Stanford] - Stanford UniversitySoundararajan, K.K.SoundararajanDepartment of Mathematics [Stanford] - Stanford UniversityRiffle shuffles of a deck with repeated cardsHAL CCSD2009card shufflinglumping of Markov chainsPoisson summation[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Episciences Iam, CoordinationKrattenthaler, Christian and Strehl, Volker and Kauers, Manuel2015-08-20 11:09:072021-02-26 10:58:012015-08-24 10:04:01enConference papershttps://hal.inria.fr/hal-01185425/document10.46298/dmtcs.2733application/pdf1We 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.