Abstract
In 2019, Ceballos and Pons introduced the s-weak order on s-decreasing trees, for any weak composition s. They proved its lattice structure and conjectured that it could be realized as the 1-skeleton of a polyhedral subdivision of a zonotope of dimension n − 1. 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, 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, 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.
| Original language | English |
|---|---|
| Article number | #60 |
| Journal | Seminaire Lotharingien de Combinatoire |
| Issue number | 91 |
| State | Published - 2024 |
Bibliographical note
Publisher Copyright:© (2024), (Universitat Wien). All rights reserved.
Funding
AHM is partially supported by NSF grant DMS-2154019. EP is supported by grants ANR-21-CE48-0020 of the French National Research Agency ANR (project PAGCAP) and PID2022-137283NB-C21 of the Spanish MCIN/AEI. DTJ is supported by grant ANR-21-CE48-0020 of the French National Research Agency ANR (project PAGCAP). MY is partially supported by Simons collaboration grant 964456. We thank V. Pons for helpful comments and for proposing this problem in the open problem session of the VIII Encuentro Colombiano de Combinatoria ECCO 2022. We also thank J. Bastidas, C. Ceballos, B. Charles, S. Giraudo, A. Padrol, V. Pilaud, G. Poullot, F. Santos, H. Thomas, Y. Vargas, the combinatorics team of LIGM, and anonymous reviewers for helpful comments and proofreading. *[email protected] †[email protected]. AHM is partially supported by NSF grant DMS-2154019. ‡[email protected]. EP is supported by grants ANR-21-CE48-0020 of the French National Research Agency ANR (project PAGCAP) and PID2022-137283NB-C21 of the Spanish MCIN/AEI. §[email protected]. DTJ is supported by grant ANR-21-CE48-0020 of the French National Research Agency ANR (project PAGCAP). ¶[email protected]. MY is partially supported by Simons collaboration grant 964456.
| Funders | Funder number |
|---|---|
| Ministerio de Ciencia, Innovación y Universidades | |
| Agencia Estatal de Investigación | |
| National Science Foundation Arctic Social Science Program | DMS-2154019, ANR-21-CE48-0020 |
| French Agence Nationale de la Recherche | PID2022-137283NB-C21 |
| Simons Foundation | 964456 |
Keywords
- Cayley trick
- flow polytope
- geometric realization
- polyhedral subdivision
- s-decreasing tree
- s-weak order
- tropical hypersurface
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics