A look at generalized perfect shuffles

TL;DR

This paper generalizes the concept of perfect shuffles by analyzing the permutation groups generated by in and out shuffles on decks split into multiple stacks, revealing that the group structure depends on specific ratios and parity, but not on the number of stacks.

Contribution

It determines the structure of permutation groups generated by generalized perfect shuffles for decks of size m^k, extending previous results to a broader class of shuffles.

Findings

Group structure depends on k/gcd(y,k) and the parity of y/gcd(y,k)

The structure is independent of the number of stacks m

Complete characterization for decks of size m^k

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 can be found for so-called generalized perfect shuffles. For these new 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 (denoted $I_m$ and $O_m$, 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 $m$.