The Dynamics and Orbit Structure of the Topdrop Map

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.