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.
We consider a generalized riffle shuffle on the colored permutation group $G_{p, n}$ and derive a determinantal formula for the probability of finding descents at given positions, proof of which is based on the bijection between the set of shuffles in question and that of non-intersecting lattice paths.
This paper concerns a probability distribution on the symmetric group generalizing the riffle shuffle of Bayer, Diaconis, and others. There are close connections with the theory of quasisymmetric and symmetric functions.
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 …
As a continuation to our previous work, we consider a generalization of carries process. Our results are : (i) right eigenvectors of the transition probability matrix, (ii) correlation of carries between different steps, and (iii) generalized riffle shuffle having the same distribution as that of the generalized carries process.
We study the cutoff phenomenon for generalized riffle shuffles where, at each step, the deck of cards is cut into a random number of packs of multinomial sizes which are then riffled together.
We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, e.g. suits disregarded or only the colors of interest. For these features, the number of shuffles drops from 3/2 log_2(n) to log_2(n). We derive closed formulae and an asymptotic `rule of thumb' formula which is remarkably accurate.
The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut 'about in half' and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function.
In the Gilbert-Shannon-Reeds shuffle, a deck of $N$ cards is cut into two approximately equal parts which are then riffled uniformly at random. Bayer and Diaconis famously showed that this Markov chain undergoes cutoff in total variation after $\frac{3\log(N)}{2 \log(2)}$ shuffles. We establish cutoff for the more general asymmetric riffle shuffles in which one cuts the deck into differently sized parts before riffling. The value of the cutoff point confirms a conjecture of Lalley from 2000. Some appealing consequences are that asymmetry always slows mixing and that total variation mixing is s…
This paper studies the game of guessing riffle-shuffled cards with complete feedback. A deck of $n$ cards labelled 1 to $n$ is riffle-shuffled once and placed on a table. A player tries to guess the cards from top and is given complete feedback after each guess. The goal is to find the guessing strategy with maximum reward (expected number of correct guesses). We give the optimal strategy for this game and prove that the maximum expected reward is $n/2+\sqrt{2/π}\cdot\sqrt{n}+O(1)$, partially solving an open problem of Bayer and Diaconis.
Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group S_n are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter $q$ satisfies $gcd(n,q-1)=1$, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. It is proved that the use of cuts does not speed up …
We study deformations of the Plancherel measure of the symmetric group by lifting them to the symmetric group and using combinatorics of card shuffling. The existing methods for analyzing deformations of Plancherel measure are not obviously applicable to the examples in this paper. The main idea of this paper is to find and analyze a formula for the total variation distance between iterations of riffle shuffles and iterations of "cut and then riffle shuffle". Similar results are given for affine shuffles, which allow us to determine their convergence rate to randomness.
The random-to-top and the riffle shuffle are two well-studied methods for shuffling a deck of cards. These correspond to the symmetric group $S_n$, i.e., the Coxeter group of type $A_{n-1}$. In this paper, we give analogous shuffles for the Coxeter groups of type $B_n$ and $D_n$. These can be interpreted as shuffles on a ``signed'' deck of cards. With these examples as motivation, we abstract the notion of a shuffle algebra which captures the connection between the algebraic structure of the shuffles and the geometry of the Coxeter groups. We also briefly discuss the generalisation to building…
In card games, in casino games with multiple decks of cards and in cryptography, one is sometimes faced with the following problem: how can a human (as opposed to a computer) shuffle a large deck of cards? The procedure we study is to break the deck into several reasonably sized piles, shuffle each thoroughly, recombine the piles, do some simple deterministic operation, for instance a cut, and repeat. This process can also be seen as a generalised Bernoulli-Laplace urn model. We use coupling arguments and spherical function theory to derive upper and bounds on the mixing times of these Markov …
In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.
The number of ``carries'' when $n$ random integers are added forms a Markov chain [23]. We show that this Markov chain has the same transition matrix as the descent process when a deck of $n$ cards is repeatedly riffle shuffled. This gives new results for the statistics of carries and shuffling.
This paper is about the following question: How many riffle shuffles mix a deck of card for games such as blackjack and bridge? An object that comes up in answering this question is the descent polynomial associated with pairs of decks, where the decks are allowed to have repeated cards. We prove that the problem of computing the descent polynomial given a pair of decks is $#P$-complete. We also prove that the coefficients of these polynomials can be approximated using the bell curve. However, as must be expected in view of the $#P$-completeness result, approximations using the bell curve are …
In 1998, Ciucu published "No-feedback card guessing for dovetail shuffles", an article which gives the optimal guessing strategy for $n$ cards ($n$ even) after $k$ riffle shuffles whenever $k>2\log_{2}\left(n\right)$. We discuss in this article the optimal guessing strategy and the asymptotic (in $n$) expected number of correct guesses for any fixed $k\geq1$. This complements the work achieved two decades ago by Ciucu.
The "carries" when n random numbers are added base b form a Markov chain with an "amazing" transition matrix determined by Holte. This same Markov chain occurs in following the number of descents or rising sequences when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra.
Consider the following one player game. A deck containing $m$ copies of $n$ different card types is shuffled uniformly at random. Each round the player tries to guess the next card in the deck, and then the card is revealed and discarded. It was shown by Diaconis, Graham, He, and Spiro that if $m$ is fixed, then the maximum expected number of correct guesses that the player can achieve is asymptotic to $H_m \log n$, where $H_m$ is the $m$th harmonic number. In this paper we consider an adversarial version of this game where a second player shuffles the deck according to some (possibly non-unif…
The Card-Cyclic-to-Random shuffle on $n$ cards is defined as follows: at time $t$ remove the card with label $t$ mod $n$ and randomly reinsert it back into the deck. Pinsky introduced this shuffle and asked how many steps are needed to mix the deck. He showed $n$ steps do not suffice. Here we show that the mixing time is on the order of $Θ(n \log n)$.
Consider an n by n array of cards shuffled in the following manner. An element x of the array is chosen uniformly at random; Then with probability 1/2 the rectangle of cards above and to the left of x is rotated 180 degrees, and with probability 1/2 the rectangle of cards below and to the right of x is rotated 180 degrees. It is shown by an eigenvalue method that the time required to approach the uniform distribution is between n^2/2 and cn^2 ln n for some constant c. On the other hand, for any k it is shown that the time needed to uniformly distribute a set of cards of size k is at most c(k)n…
This paper considers the effect of riffle shuffling on decks of cards, allowing for some cards to be indistinguishable from other cards. The dual problem of dealing a game with hands, such as bridge or poker, is also considered. The Gilbert-Shannon-Reeds model of card shuffling is used, along with variation distance for measuring how close to uniform a deck has become. The surprising results are that for a deck with only two types of cards (such as red and black), the shuffler can greatly improve the randomness of the deck by insuring that the top and bottom cards are the same before shuffling…
The card-cyclic-to-random shuffle is the card shuffle where the $n$ cards are labeled $1,\ldots,n$ according to their starting positions. Then the cards are mixed by first picking card $1$ from the deck and reinserting it at a uniformly random position, then repeating for card $2$, then for card $3$ and so on until all cards have been reinserted in this way. Then the procedure starts over again, by first picking the card with label $1$ and reinserting, and so on. Morris, Ning and Peres \cite{MNP} recently showed that the order of the number of shuffles needed to mix the deck in this way is $n\…
Consider a permutation $σ\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $σ_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We study some probabilistic and combinatorial aspects of the shuffle on $S_n$ defined by removing and then randomly reinserting each of the $n$ cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be q…
The Thorp shuffle is defined as follows. Cut the deck into two equal piles. Drop the first card from the left pile or the right pile according to the outcome of a fair coin flip; then drop from the other pile. Continue this way until both piles are empty. We show that the mixing time for the Thorp shuffle with $2^d$ cards is polynomial in $d$.
In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of $N$ cards. We prove that around time $N^2\log N/(2π^2)$, the total variation distance to equilibrium of the deck distribution drops abruptly from $1$ to $0$, and that the separation distance has a similar behavior but with a transition occurring at time $(N^2\log N)/π^2$. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length $N$ with $k$ particles.
E. Thorp introduced the following card shuffling model. Suppose the number of cards $n$ is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We show that if $n$ is a power of 2 then the mixing time of the Thorp shuffle is $O(\log^3 n)$. Previously, the best known bound was $O(\log^4 n)$.
Consider a permutation $σ\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $σ_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We define two cyclic to random transposition shuffles. The first one works as follows: for $j=1,..., n$, on the $j$-th step transpose the card that was \it originally\rm\ the $j$-th from the left with a random card (possibly itself). The second shuffle works as follows: on the $j$-th step, transpose the card that is \it currently\rm\ in the $j$-th position from the left with a random card (possibly itself…
The game of memory is played with a deck of n pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one card then another. The cards are removed from play if they match. Otherwise, they are flipped back over and the next move commences. A game ends when all pairs have been matched. We determine that, when the game is played optimally, as n tends to infinity: 1) The expected number of moves is (3 - 2 ln 2)n + 7/8 - 2 ln 2 (approximately 1.61 n), 2) The expected number of times two matching cards are unwitt…
We consider the following game that has been used as a way of testing claims of extrasensory perception (ESP). One is given a deck of $mn$ cards comprised of $n$ distinct types each of which appears exactly $m$ times: this deck is shuffled and then cards are discarded from the deck one at a time from top to bottom. At each step, a player (whose psychic powers are being tested) tries to guess the type of the card currently on top, which is then revealed to the player before being discarded. We study the expected number $S_{n,m}$ of correct predictions a player can make: one could always guess t…
A deck of $n$ cards is shuffled by repeatedly moving the top card to one of the bottom $k_n$ positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as $k_n$ ranges from a constant to $n$. We also consider a symmetric variant of this shuffle in which at each step either the top card is randomly inserted into the bottom $k_n$ positions or a random card from the bottom $k_n$ positions is moved to the top. For this reversible shuffle we derive bounds on the $L^2$ mixing time. Finally, we transfer mixing time estimates for the above shuffle…
In a recent work Conger and Howald derived asymptotic formulas for the randomness, after shuffling, of decks with repeating cards or all-distinct decks dealt into hands. In the latter case the deck does not need to be fully randomized: the order of cards received by a player is indifferent. They called these cases the "fixed source" and the "fixed target" case, respectively, and treated them separately. We build on their results and mix these two cases: we obtain asymptotic formulas for the randomness of a deck of repeating cards which is shuffled and then dealt into hands of players. We confi…
We consider the following game. A deck with $m$ copies of each of $n$ distinct cards is shuffled in a perfectly random way. The Guesser sequentially guesses the card from top to bottom. After each guess, the Guesser is informed whether the guess is correct. The goal is to maximize the expected number of correct guesses. We prove that, if $n= Ω(\sqrt{m})$, then at most $m+O(\sqrt{m})$ cards can be guessed correctly. Our result matches a lower bound of the maximal expected payoff by Diaconis, Graham and Spiro when $n=Ω(m)$.
The overhand shuffle is one of the ``real'' card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37--49] who showed that the mixing time with respect to variation distance is at least of order $n^2$ and at most of order $n^2\log n$. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274--325] to establish a lower bound of order $n^2\log n$, thereby showing that $n^2\log n$ is indeed the correct order of the mixing time. It is our …
We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models. E. Thorp introduced the following card shuffling model in 1973. Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log^4 n). Previously, the best …
The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either the $n$th card or the $(n-k)$th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of $k$ and $n$, has surprising behavior. For example, suppose $k$ is the closest integer to $αn$ for a fixed real $α\in(0,1)$. Then for rational $α$ the spectral gap is $Θ(n^{-2})$, while for poorly approximable irrational numbers $α$, such as the reciprocal of the golden ratio, the spectral gap is $Θ(n^{-3/2})$.
We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate Cn(log{n}) with a moderate constant, C. This rate was determined by Diaconis and Shahshahani, but the question of a natural probabilistic coupling proof has been missing, and questions of its existence have been raised. The proof, and indeed any proof, requires that we enlarge the methodology of coupling to include intuitive but non-adapted coupling rules, because a typical Markovian coupling is incapable of resolving finer questions of rates.
Consider the interchange process on a connected graph $G=(V,E)$ on $n$ vertices. I.e.\ shuffle a deck of cards by first placing one card at each vertex of $G$ in a fixed order and then at each tick of the clock, picking an edge uniformly at random and switching the two cards at the end vertices of the edge with probability 1/2. Well known special cases are the random transpositions shuffle, where $G$ is the complete graph, and the transposing neighbors shuffle, where $G$ is the $n$-path. Other cases that have been studied are the $d$-dimensional grid, the hypercube, lollipop graphs and Erd\H o…
Consider the following method of card shuffling. Start with a deck of $N$ cards numbered 1 through N. Fix a parameter $p$ between 0 and 1. In this model a ``shuffle'' consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads then we arrange the two cards so that the lower numbered card comes before the higher numbered card. If the coin comes up tails then we arrange the cards with the higher numbered card first. In this paper we prove that for all p not equal to 1/2, the mixing time of this card shuffling is O…
