2023/07/07 by Rafael S. González D’León, Alejandro H. Morales, D'León, Rafael S. González +7 · 1 citation
Mathematics · #06B05 #52B22 #52C07 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #G.2.1 #Primary 52B12 #Secondary 05C21
paper · pdf · doi:10.48550/arxiv.2307.03474
openalex publication_date 2023/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.