card shuffling

We investigate a question posed by Diaconis & Fulman (2023) regarding modular in-shuffling and the distribution of card positions in binary form. Specifically, we consider the sequence 𝑋 𝑘 , which records whether the position of a tracked card lies in the top half of a deck after 𝑘 iterations of the modular in-shuffle map 𝐼(𝑗) = 2𝑗 mod 𝑝, where 𝑝 is a is a prime for which 2 is a primitive root. We prove that the random sequence {𝑋 𝑘 } is equidistributed and asymptotically behaves like a sequence of independent and identically distributed Bernoulli(1 2) random variables. We provide both theoretical justification and empirical evidence with computed values of 𝑋 𝑘 for large 𝑝 supporting this behavior.

The following paper focuses on the role and function of symbols in magical practice. The first part explains the symbol as a semiotic term with all its fundamental features, including the necessary function and condition of ordinary communication. The next section deals with symbols that fulfill the essential role of transforming reality within the framework of magical practice. Examples of sigil magic, Tarot, and amulets explain the paradigm, which uses symbolic language as a mandatory condition for its existence and simultaneously ignores the shared awareness of the origin of the symbol as a constitutional element of its functionality. This paper demonstrates how the symbols used in everyday communication and the symbols used in magical practice are two different entities performing distinct kinds of function, and primarily that they differ in requirements for being created and functional.

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 walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate.

A theoretical result by Ramnath and Scully (1996) provides a detailed proof for the correctness of a computationally efficient procedure for moving a card from any position i to any position j in a deck of 2n cards using only consecutive perfect in-shuffles and outshuffles. The following addendum to their paper provides a brief overview of this procedure as well as a program written in Python for implementing their result and computing the shortest sequence of in-shuffles and out-shuffles necessary to move a card from any position to any position. For a review of the underlying jargon and motivation for the problem, refer to [1] or [2].

The following paper focuses on the role and function of symbols in magical practice. The first part explains the symbol as a semiotic term with all its fundamental features, including the necessary function and condition of ordinary communication. The next section deals with symbols that fulfill the essential role of transforming reality within the framework of magical practice. Examples of sigil magic, Tarot, and amulets explain the paradigm, which uses symbolic language as a mandatory condition for its existence and simultaneously ignores the shared awareness of the origin of the symbol as a constitutional element of its functionality. This paper demonstrates how the symbols used in everyday communication and the symbols used in magical practice are two different entities performing distinct kinds of function, and primarily that they differ in requirements for being created and functional.

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 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 n, thereby showing that n^2 n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n^3 n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77--85] for the Rudvalis shuffle.

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 Erdős-Rényi random graphs above the threshold for connectedness. In this paper the problem is studied for general G. Special attention is focused on trees, random trees and the giant component of critical and supercritical G(N, p) random graphs. Upper and lower bounds on the mixing time are given. In many of the cases, we establish the exact order of the mixing time. We also mention the cases when G is the hypercube and when G is a bounded-degree expander, giving upper and lower bounds on the mixing time.

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 hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the (n 3 log n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77-85] for the Rudvalis shuffle.

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

In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into two packets and then the two packets are interleaved (e.g. by moving cards one at a time from the bottom of one or the other of the packets to the top of the sorted deck). As a special case of this, a (p,q)-shuffle, for numbers p and q with p + q = n, is a riffle in which the first packet has p cards and the second packet has q cards. [1]

For positive integers k and n the group of perfect k-shuffles with a deck of kn cards is a subgroup of the symmetric group S kn . The structure of these groups was found for k = 2 by Diaconis, Graham, and Kantor and for k ≥ 3 and a deck of k m cards by Medvedoff and Morrison. They also conjectured that for k = 4 and deck size 2 m , m odd, the group is isomorphic to the group of affine transformations of an m-dimensional vector space over the field of order 2. That conjecture is proved in this paper, and a complete conjecture is stated for the the structure of the shuffle groups for all k and n.