Shuffling with ordered cards

TL;DR

This paper analyzes a specific card shuffling method involving ordered decks, identifying invariant and periodic configurations, and characterizing possible periods based on number-theoretic conditions.

Contribution

It introduces a mathematical framework for understanding the invariance and periodicity in a particular ordered card shuffling process, including conditions for possible periods.

Findings

Identified invariant and periodic stacks under the shuffling process.

Derived conditions for the periods based on gcd and order functions.

Characterized all possible periods when gcd(q,k)=1.

Abstract

We consider a problem of shuffling a deck of cards with ordered labels. Namely we split the deck of N=k^tq cards (where t>=1 is maximal) into k equally sized stacks and then take the top card off of each stack and sort them by the order of their labels and add them to the shuffled stack. We show how to find stacks of cards invariant and periodic under the shuffling. We also show when gcd(q,k)=1 the possible periods of this shuffling are all divisors of order_k(N-q).