Realizing the 𝑠-permutahedron via flow polytopes

Ceballos and Pons introduced the s s -weak order on s s -decreasing trees, for any weak composition s s. They proved that it has a lattice structure and further conjectured that it can be realized as the 1 1 -skeleton of a polyhedral subdivision of a polytope. We answer their conjecture in the case where s s is a strict composition by providing three geometric realizations of the s s -permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second one, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third one, obtained using tropical geometry, is the 1 1 -skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured.