Related papers: A Random Card Shuffling Process

We consider a card guessing strategy for a stack of cards with two different types of cards, say $m_1$ cards of type red (heart or diamond) and $m_2$ cards of type black (clubs or spades). Given a deck of $M=m_1+m_2$ cards, we propose a…

Standard perfect shuffles involve splitting a deck of $2n$ cards into two stacks and interlacing the cards from the stacks. There are two ways that this interlacing can be done, commonly referred to as an in shuffle and an out shuffle,…

A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the…

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…

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 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…

Consider a card guessing game with complete feedback in which a deck of $n$ cards ordered $1,\dots, n$ is riffle-shuffled once. With the goal to maximize the number of correct guesses, a player guesses cards from the top of the deck one at…

We introduce a new type of card shuffle called one-sided transpositions. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric…

Frequently, randomly organized data is needed to avoid an anomalous operation of other algorithms and computational processes. An analogy is that a deck of cards is ordered within the pack, but before a game of poker or solitaire the deck…

We analyze the mixing time of a popular shuffling machine known as the shelf shuffler. It is a modified version of a $2m$-handed riffle shuffle ($m=10$ in casinos) in which a deck of $n$ cards is split multinomially into $2m$ piles, the…

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…

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…

In this expository article, we highlight the direct connection between card shuffling and the functions known as $P$-partitions that come from algebraic combinatorics. While many (but not all) of the results we discuss are known, we give a…

We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is "reversed", and then the cards are interlaced. Flip shuffles are when the reversal comes from flipping the half over so that we also need to…

In card-based cryptography, a deck of physical cards is used to achieve secure computation. A shuffle, which randomly permutes a card-sequence along with some probability distribution, ensures the security of a card-based protocol. The…

A pile-scramble shuffle is one of the most effective shuffles in card-based cryptography. Indeed, many card-based protocols are constructed from pile-scramble shuffles. This article aims to study the power of pile-scramble shuffles. In…

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…

Various card tricks involve under-down dealing, where alternatively one card is placed under the deck and the next card is dealt. We study how the cards need to be prepared in the deck to be dealt in order. The order in which the $N$ cards…

We investigate the $k$-cycle shuffle on repeated cards, namely on a deck consisting of $l$ identical copies of each of $m$ card types, with total size $n=ml$. We establish asymptotic results for the total variation mixing of this shuffle,…

Consider a uniformly random deck consisting of cards labelled by numbers from $1$ through $n$, possibly with repeats. A guesser guesses the top card, after which it is revealed and removed and the game continues. What is the expected number…