Vol. 14, No. 5, 2021

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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
Abstract

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, respectively. In 1983, Diaconis, Graham, and Kantor determined the permutation group generated by in and out shuffles on a deck of 2n 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 mn 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 mk cards, together with my-shuffles, for all possible values of m, k, and y. The group structure is completely determined by kgcd(y,k) and the parity of ygcd(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