Year after year, I accumulate computational results of several kind: here there are! The data are not perfectly sorted, and, sadly, I did not have time to do better. As they might be usefull for your research, please feel free to download and play with them.
I computed "all" the faces of the submodular cone for n = 3 and n = 4, and, with the help of Winfried Bruns, some of the faces for n = 5, and some rays for n = 6. Each face of the submodular cone is associated to (a class of normally equivalent) polytopes, which are called generalized permutahedra, a.k.a deformed permutahedra (one can also associate the faces to classes of submodular functions, among others: I made a choice). I tried to make the data understandable, but if you are totally unfamiliar with this subject, you will probably need to read some papers first: knowing the definition of a generalized permutahedra and of the submodular cone is necessary; knowing what a deformation and a deformation cone are is recommanded.
The folder GP_n=4_up_to_symmetries contains a READ_ME.txt file that details the above short explanation: you should read it! See also the slides 28 to 33 of my talk on the submodular cone for some drawings and explanations.
Here are the links to the data:
| n | Data | Description |
|---|---|---|
| 3 | GP_n=3_up_to_symmetries | All faces of the submodular cone for n = 3, up to: central symmetry, permutation of coordinates (and normal equivalence) |
| 4 | GP_n=4_up_to_symmetries | All faces of the submodular cone for n = 4, up to: central symmetry, permutation of coordinates (and normal equivalence) |
| 5 | GP_n=5_up_to_symmetries (ask by email) | More than 6 600 000 faces of the submodular cone for n = 5, up to: central symmetry, permutation of coordinates (and normal equivalence) |
| 6 | GP_n=6_up_to_symmetries (ask by email) | 126 629 rays of the submodular cone for n = 6, up to: central symmetry, permutation of coordinates (and normal equivalence) |
| 3 | All_GP_n=3 | All generalized permutahedra for n = 3 (up to normal equivalence only) |
| 4 | All_GP_n=4 | All generalized permutahedra for n = 4 (up to normal equivalence only) |
For each graph G = (V, E) (with up to 6 vertices), I computed the polytope algebra and the weight algebra of its graphical zonotope Z_G (please, find the definitions in the literature). I recorded the dimension of each graded piece of these algebras, and compare with the h-vector (the usual polynomial transform of the f-vector), and the possible in-degree vectors of Z_G.
As per usual, reading the READ_ME.txt file is not mandatory but quite usefull! Even though it is a data-base that you can open with a computer (using, e.g., Python + SageMath), the data are also meant to be human-readable.
Here are the links to the data: Polytope_algebra_graphical_zonotopes.