Random for a thorough treatment of such results see. longest cycle, descent structure, etc, to become Many results are known for how long it takes certain features of a Shuffles are required in these situations, and so we consider a deck to It is natural, therefore, to ask how many Guessing experiments, a Zener deck of 25 cards with each of 5 symbols For instance, in Baccarat, suits are irrelevantĪnd all 10's and picture cards are equivalent, and in ESP card In many card games, only certain aspects of Results, however, look at all aspects of a permutation, i.e. Shuffles are necessary and sufficient to make separation small. Make the total variation distance small, while 2(n) + c That 3/2 (n) + c shuffles are necessary and sufficient to In widely cited works, Aldous and Bayer and Diaconis show Which makes separation a good measure to use when total variation From the formulas above, one canĮasily see that separation provides an upper bound for total variation, Here, only a single probability needs to be computed, though as we Therefore we will also consider the separation distance defined by In general, the formulas for () may be quiteĬomplicated, making calculations of total variation intractable. The total variation distance is defined by The purposes of this paper we restrict our attention to total variation Several ways to measure the distance between and U, though for ForĪ deck with n distinct cards, U = 1/n!, and for a more general deck with To that end, denote the uniform distribution by U = U(). Thus it is enough to study a single a-shuffle of the deck. General a-shuffles is as nice as possible, namely () denote this measure, they show that convolution of Packets by multinomial distribution and cards are successively droppedįrom packets with probability proportional to packet size. Natural extension to shuffling with a hands: the deck is cut into a This shuffling model, which accurately models how most peopleĪctually shuffle a deck of cards, was introduced by Gilbert and Shannonīayer and Diaconis generalized this to a-shuffles, which is the Successively dropping cards from either pile with probability To the binomial distribution, and then riffling the piles together by
We shuffle the deck by first cutting it into two piles according To be precise, we consider a 'deck' to be a multiset of nĬards. Properties of the deck that are of interest.
We use to measure distance to uniformity, but also on the particular Our answer depends not only on the metric Gilbert-Shannon-Reeds model for riffle shuffling a deck of nĬards and ask how many times the deck must be shuffled for the cards toīe in close to random order.
In this paper we study the mixing properties of the Retrieved from Ī basic question in scientific computing is 'How many times APA style: Riffle shuffles of a deck with repeated cards.Riffle shuffles of a deck with repeated cards." Retrieved from
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