The Annals of Probability

Riffle shuffles of decks with repeated cards

Mark Conger and D. Viswanath

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By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11—with the former number applying when the metric used to define mixing is the total variation distance and the latter when it is the separation distance. This and other related results assume all 52 cards in the deck to be distinct and require all 52! permutations of the deck to be almost equally likely for the deck to be considered well mixed. In many instances, not all cards in the deck are distinct and only the sets of cards dealt out to players, and not the order in which they are dealt out to each player, needs to be random. We derive transition probabilities under riffle shuffles between decks with repeated cards to cover some instances of the type just described. We focus on decks with cards all of which are labeled either 1 or 2 and describe the consequences of having a symmetric starting deck of the form 1,…,1,2,…,2 or 1,2,…,1,2. Finally, we consider mixing times for common card games.

Article information

Ann. Probab., Volume 34, Number 2 (2006), 804-819.

First available in Project Euclid: 9 May 2006

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Descents multisets riffle shuffles


Conger, Mark; Viswanath, D. Riffle shuffles of decks with repeated cards. Ann. Probab. 34 (2006), no. 2, 804--819. doi:10.1214/009117905000000675.

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