The Dynamics and Orbit Structure of the Topdrop Map
Nathan R. Krause
TL;DR
The paper studies the topdrop map $T$ on the symmetric group $S_n$, proving bijectivity and introducing topdrop-necklaces to classify orbits. It develops a necklace-based orbit-counting framework and provides exact counts for small orbit sizes, along with lower bounds for larger sizes, complemented by parity symmetries that constrain valid necklaces. The main contributions are the inverse relation $T^{-1}(\pi)=\mathrm{rev}(T(\mathrm{rev}(\pi)))$, the necklace-based orbit counting formula, and the detailed enumeration results up to $S_{14}$, establishing both exact and asymptotic-type bounds. The results offer a structured combinatorial approach to permutation dynamics under the topdrop map and pave the way for computational exploration of longer orbits and richer necklace types, with potential connections to topswops variants and broader permutation-dynamics questions.
Abstract
We study the topdrop map, a mapping on permutations in $S_n$ related to card shuffling. We show this map is bijective and study its orbit structure. We introduce the notion of the topdrop-necklace as a way of classifying the orbits of the map and establish a general theorem to count orbits using topdrop-necklaces. We then provide exact counts for orbits of size two through five and lower bounds for the number of orbits of sizes six and eight. We show symmetries in orbits which happen when $n$ or $n-1$ is in the topdrop-necklace, count these orbits, and show that they have even size. We prove a restriction on topdrop-necklaces based on permutation parity.