Ceballos and Pons introduced the s-weak order on s-decreasing trees, for
any weak composition s. They proved that it has a lattice structure and
further conjectured that it can be realized as the 1-skeleton of a polyhedral
subdivision of a polytope. We answer their conjecture in the case where s is
a strict composition by providing three geometric realizations of the
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-skeleton of a polyhedral complex for which we can provide explicit
coordinates of the vertices and whose support is a permutahedron as
conjectured.Comment: 39 pages, 14 figure