A look at generalized perfect shuffles

Johnson, Samuel; Manny, Lakshman; Van Cott, Cornelia A.; Zhang, Qiyu

doi:10.2140/involve.2021.14.813

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A look at generalized perfect shuffles

Samuel Johnson, Lakshman Manny, Cornelia A. Van Cott and Qiyu Zhang

Vol. 14 (2021), No. 5, 813–828

DOI: 10.2140/involve.2021.14.813

Abstract

Standard perfect shuffles involve splitting a deck of $2 n$ 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, respectively. In 1983, Diaconis, Graham, and Kantor determined the permutation group generated by in and out shuffles on a deck of $2 n$ cards for all $n$ . Diaconis et al. concluded their work by asking whether similar results hold for so-called generalized perfect shuffles. For these shuffles, we split a deck of $m n$ cards into $m$ stacks and similarly interlace the cards with an in $m$ -shuffle or out $m$ -shuffle, respectively. In this paper, we find the structure of the group generated by these two shuffles for a deck of $m^{k}$ cards, together with $m^{y}$ -shuffles, for all possible values of $m$ , $k$ , and $y$ . The group structure is completely determined by $k ∕ gcd (y, k)$ and the parity of $y ∕ gcd (y, k)$ . In particular, the group structure is independent of the value of $m$ .

I must complain the cards are ill shuffled till I have a good hand. — Jonathan Swift (1667–1745)

Keywords

perfect shuffle, permutation group, symmetric group

Mathematical Subject Classification

Primary: 05A05, 20B35

Milestones

Received: 19 January 2021

Revised: 14 June 2021

Accepted: 14 June 2021

Published: 9 February 2022

Communicated by Joseph Gallian

Authors

Samuel Johnson
Department of Mathematics and Statistics University of California Santa Cruz, CA United States

Lakshman Manny
Department of Mathematics and Statistics University of San Francisco San Francisco, CA United States

Cornelia A. Van Cott
Department of Mathematics and Statistics University of San Francisco San Francisco, CA United States

Qiyu Zhang
Department of Mathematics and Statistics University of San Francisco San Francisco, CA United States

Vol. 14, No. 5, 2021