Polyurethane toggles (Q2188837)

Summary: We consider the involutions known as \textit{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 \textit{M. Bouvel} and \textit{O. Guibert} [Ann. Comb. 18, No. 2, 199--232 (2014; Zbl 1295.05009)] and also generalizes a theorem of \textit{M. Bousquet-Mélou} [Discrete Math. 225, No. 1--3, 25--50 (2000; Zbl 0961.05001)]. We also settle a conjecture of Bouvel and Guibert [loc. cit.]. 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.