Related papers: The $S_k$ shuffle block dynamics
We investigate the mixing properties of a model of reversible Markov chains in random environment, which notably contains the simple random walk on the superposition of a deterministic graph and a second graph whose vertex set has been…
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…
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 study Markov chains which model genome rearrangements. These models are useful for studying the equilibrium distribution of chromosomal lengths, and are used in methods for estimating genomic distances. The primary Markov chain studied…
We develop a combinatorial and order-theoretic framework for shuffles, understood as ordered concatenations of indexed families of sequences that induce total orders on the natural numbers. Motivated by the classical \v{S}arkovski\u{i}…
Relying on the optimal guessing strategy recently found for a no-feedback card guessing game with $k$-time riffle shuffles, we derive an exact, closed-form formula for the expected number of correct guesses and higher moments for a $1$-time…
In this paper, we present a detailed proof for the exhibition of a cutoff for the one-sided transposition (OST) shuffle on the generalized symmetric group $G_{m,n}$. Our work shows that based on techniques for $m \leq 2$ proven by…
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…
We introduce a multi-colour multi-urn generalisation of the Bernoulli-Laplace urn model, consisting of $d$ urns, $m$ colours, and $dmn$ balls, with $dn$ balls of each colour and $mn$ balls in each urn. At each step, one ball is drawn…
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…
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…
We study the mixing time of the $(n,k)$ Bernoulli--Laplace urn model, where $k\in\{0,1,\ldots,n\}$. Consider two urns, each containing $n$ balls, so that when combined they have precisely $n$ red balls and $n$ white balls. At each step of…
Inspired by a common technique for shuffling a deck of cards on a table without riffling, we formalize the pile shuffle and investigate its capabilities as a sorting device. Our study is novel in that we consider pile shuffle in three…
Analogues of 1-shuffle elements for complex reflection groups of type $G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms of rotational permutations of polygonal cards is given. We compute the eigenvalues, and…
We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results.…
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…
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,…
We consider an ordinary differential equation with a unique hyperbolic attractor at the origin, to which we add a small random perturbation. It is known that under general conditions, the solution of this stochastic differential equation…
Let $B_t(n)$ be the number of set partitions of a set of size~$t$ into at most $n$ parts and let $B'_t(n)$ be the number of set partitions of $\{1,\ldots, t\}$ into at most $n$ parts such that no part contains both $1$ and~$t$ or both $i$…
Consider a random permutation of $kn$ objects that permutes $n$ disjoint blocks of size $k$ and then permutes elements within each block. Normalizing its cycle lengths by $kn$ gives a random partition of unity, and we derive the limit law…