Polyurethane Toggles

TL;DR

This paper explores involutions called toggles that generate a group acting on permutations, providing new proofs and generalizations of known theorems related to stack-sorting and permutation properties.

Contribution

It introduces a group generated by toggle involutions acting on permutations, characterizes its orbits, and applies this to simplify proofs and extend results in stack-sorting theory.

Findings

Characterized orbits of the toggle group in terms of permutation skeletons.

Provided a simple proof of a generalized theorem related to stack-sorting.

Settled a conjecture of Bouvel and Guibert and connected toggle actions to permutation group structure.

Abstract

We consider the involutions known as "toggles," which have been used to give simplified proofs of the fundamental properties of the promotion and evacuation maps. We transfer these involutions so that they generate a group $\mathscr P_n$ that acts on the set $S_n$ of permutations of $\{1,\ldots,n\}$. After characterizing its orbits in terms of permutation skeletons, we apply the action in order to understand West's stack-sorting map. We obtain a very simple proof of a result that clarifies and extensively generalizes a theorem of Bouvel and Guibert and also generalizes a theorem of Bousquet-M\'elou. We also settle a conjecture of Bouvel and Guibert. We prove a result related to the recently-introduced notion of postorder Wilf equivalence. Finally, we investigate an interesting connection among the action of $\mathscr P_n$ on $S_n$, the group structure of $S_n$, and the stack-sorting map.