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arXiv · 2410.22548

Fixed Point Homing Shuffles

Jonathan Parlett
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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.

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Jonathan Parlett. 2025-07-18. Fixed Point Homing Shuffles. https://doi.org/10.46298/dmtcs.14653

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