Related papers: Top to random shuffles on colored permutations

Let a deck of n cards be shuffled by successively exchanging the cards in positions 1, 2, ..., n with cards in randomly chosen positions. We show that for n equal to 18 or greater, the identity permutation is the most likely. We prove a…

We consider a Markov chain on invertible $n\times n$ matrices with entries in $\mathbb{Z}_2$ which moves by picking an ordered pair of distinct rows and add the first one to the other, modulo $2$. We establish a logarithmic Sobolev…

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…

Consider shuffling a deck of $n$ cards, labeled $1$ through $n$, as follows: at each time step, pick one card uniformly with your right hand and another card, independently and uniformly with your left hand; then swap the cards. How long…

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…

Let $P$ be a bistochastic matrix of size $n$, and let $\Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=P\Pi$. In other words, the chain…

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…

Establishing cutoff, an abrupt transition from "not mixed" to "well mixed", is a classical topic in the theory of mixing times for Markov chains. Interest has grown recently in determining not only the existence of cutoff and the order of…

In this paper, we investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of…

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.…

We consider a family of card shuffles of $n$ cards in which the allowed moves involve transpositions corresponding to the Jucys--Murphy elements of the symmetric group $\{S_m\}_{m \leq n}$. We determine the eigenvalues of the corresponding…

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…

In this paper, we study the biased random transposition shuffle, a natural generalization of the classical random transposition shuffle studied by Diaconis and Shahshahani. We diagonalize the transition matrix of the shuffle and use these…

The best known lower and upper bounds on the mixing time for the random-to-random insertions shuffle are $(1/2-o(1))n\log n$ and $(2+o(1))n\log n$. A long standing open problem is to prove that the mixing time exhibits a cutoff. In…

We develop a general theory for Markov chains whose transition probabilities are the coefficients of descent operators on combinatorial Hopf algebras. These model the breaking-then-recombining of combinational objects. Examples include the…

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,…

In each step of the overlapping cycles shuffle on $n$ cards, a fair coin is flipped which determines whether the $m$th card or the $n$th card is moved to the top of the deck. Angel, Peres, and Wilson showed the following interesting fact:…

We introduce and analyze the $S_k$ shuffle on $N$ cards, a natural generalization of the celebrated random adjacent transposition shuffle. In the $S_k$ shuffle, we choose uniformly at random a block of $k$ consecutive cards, and shuffle…

In this paper we study the mixing time of a biased transpositions shuffle on a set of $N$ cards with $N/2$ cards of two types. For a parameter $0<a \le 1$, one type of card is chosen to transpose with a bias of $\frac{a}{N}$ and the other…

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…