Matroid 4.6.0
| Label | 4.6.0 |
|---|---|
| Id | r4_n6_*************** |
| Rank | 4 |
| n | 6 |
Basic invariants
Representability
| Characteristic set | characteristic 0 and all primes [0] |
|---|---|
| Realizable | yes |
| Realizable char0 | yes |
| Regular | no |
| Binary | no |
| Ternary | no |
| Quaternary | no |
| Graphic | no |
Realization space
| Status | computed_realization_space |
|---|---|
| Dimension of realization space | 3 |
| Expected dimension over ℤ | not computed |
| Components of realization space | 1 |
| Free realization space | yes |
| Principal ideal | yes |
| Birational type | rational how determined: birational type method: free_presentation |
| Birational type components | show[
{
"dim": 3,
"free_rank": 3,
"torus_rank": 3,
"core_dim": 0,
"qbar_components": 1,
"type": "rational",
"reason": "free realization space"
}
] |
| Good basis | not computed |
| Good basis v2 | 1,2,3,4 |
Geometry
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | 1 |
Tutte polynomial
\(T = x^{4} + 2 x^{3} + 3 x^{2} + 4 x + y^{2} + 4 y\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}, x_{2}, x_{3}]\) |
|---|---|
| Defining ideal | \(\left(0\right)\) |
| Inequations | \(x_{1} - x_{2} \neq 0\), \(x_{2} - 1 \neq 0\), \(x_{1} - 1 \neq 0\), \(x_{1} - x_{3} \neq 0\), \(x_{3} - 1 \neq 0\), \(x_{2} - x_{3} \neq 0\), \(x_{1} \neq 0\), \(x_{2} \neq 0\), \(x_{3} \neq 0\) |
| Realization matrix | \(\begin{pmatrix}1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & x_{1} \\ 0 & 0 & 1 & 0 & 1 & x_{2} \\ 0 & 0 & 0 & 1 & 1 & x_{3}\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (15)
{1,2,3,4} {1,2,3,5} {1,2,4,5} {1,3,4,5} {2,3,4,5} {1,2,3,6} {1,2,4,6} {1,3,4,6} {2,3,4,6} {1,2,5,6} {1,3,5,6} {2,3,5,6} {1,4,5,6} {2,4,5,6} {3,4,5,6}Non-bases (0)
Circuits (6)
{1,2,3,4,5} {1,2,3,4,6} {1,2,3,5,6} {1,2,4,5,6} {1,3,4,5,6} {2,3,4,5,6}Flats by rank (43)
- rank 0 (1): {}
- rank 1 (6): {1} {2} {3} {4} {5} {6}
- rank 2 (15): {1,2} {1,3} {2,3} {1,4} {2,4} {3,4} {1,5} {2,5} {3,5} {4,5} {1,6} {2,6} {3,6} {4,6} {5,6}
- rank 3 (20): {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,5} {1,3,5} {2,3,5} {1,4,5} {2,4,5} {3,4,5} {1,2,6} {1,3,6} {2,3,6} {1,4,6} {2,4,6} {3,4,6} {1,5,6} {2,5,6} {3,5,6} {4,5,6}
- rank 4 (1): {1,2,3,4,5,6}
Hyperplanes (20)
{1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,5} {1,3,5} {2,3,5} {1,4,5} {2,4,5} {3,4,5} {1,2,6} {1,3,6} {2,3,6} {1,4,6} {2,4,6} {3,4,6} {1,5,6} {2,5,6} {3,5,6} {4,5,6}Dual (rank 2)
Revlex encoding in this labeling (not canonicalized, so not linked): ***************
Bases: {5,6} {4,6} {3,6} {2,6} {1,6} {4,5} {3,5} {2,5} {1,5} {3,4} {2,4} {1,4} {2,3} {1,3} {1,2}
Loops: none. Parallel classes of size > 1: none.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("***************", 4, 6)