Polyurethane toggles

Colin Defant¹

¹Princeton University

The electronic journal of combinatorics, Tome 27 (2020) no. 2

Résumé

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.

Détail

DOI : 10.37236/9097

Classification : 05A05, 05E10, 05E18, 05A19
Mots-clés : polyurethane group, rooted plane trees