Fixed Point Homing Shuffles

TL;DR

This paper introduces fixed point homing shuffles, a family of permutation maps that generalize known shuffling problems, proving convergence of their iterates and characterizing unsortable permutations.

Contribution

It defines fixed point homing shuffles, proves their convergence, characterizes permutations they cannot sort, and analyzes the sorting process's worst-case iterations.

Findings

Iterates of fixed point homing shuffles always converge.

Characterization of permutations that cannot be sorted by these shuffles.

Analysis of the maximum number of iterations needed for sorting in the worst case.

Abstract

We study a family of maps from $S_n \to S_n$ we call fixed point homing shuffles. These maps generalize a few known problems such as Conway's Topswops, and a card shuffling process studied by Gweneth McKinley. We show that the iterates of these homing shuffles always converge, and characterize the set $U_n$ of permutations that no homing shuffle sorts. We also study a homing shuffle that sorts anything not in $U_n$, and find how many iterations it takes to converge in the worst case.