PermutahedronVertex from Permutation
The vertex (σ(1), …, σ(n)), or its Loday point, is an ordered tuple of coordinates, while finset is an unordered set of positions (the carrier DescentSet and PeakSet use). Reordering the coordinates gives a different vertex but the same finset, so the map would not be injective. A finset-valued PermutahedronVertex is a type mismatch, not a gap to close.
Named and typed, but not defined yet.
Called as CombinatorialMap(x, "PermutahedronVertex"). The head's page has its examples.