Matroid 4.9.5
| Label | 4.9.5 |
|---|---|
| Id | r4_n9_0********0******************************************************************************************************************** |
| Rank | 4 |
| n | 9 |
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 | 2 |
| Expected dimension over ℤ | not computed |
| Components of realization space | not computed |
| Free realization space | yes |
| Principal ideal | yes |
| Birational type | rational how determined: birational type method: free_presentation |
| Birational type components | show[
{
"dim": 10,
"free_rank": 10,
"torus_rank": 10,
"core_dim": 0,
"qbar_components": 1,
"type": "rational",
"reason": "free realization space"
}
] |
| Good basis | not computed |
| Good basis v2 | 1,2,3,5 |
Geometry
Other
| Char poly splits | false |
|---|---|
| Simplicial arrangement | no |
| Realization space: n qbar components | not computed |
Tutte polynomial
\(T = x_{0}^{4} + 5 x_{0}^{3} + 15 x_{0}^{2} + 2 x_{0} x_{1} + 33 x_{0} + x_{1}^{5} + 4 x_{1}^{4} + 10 x_{1}^{3} + 20 x_{1}^{2} + 33 x_{1}\)
Realization space
| Ring | \(\mathbb{Z}[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}]\) |
|---|---|
| Defining ideal | \(\left(0\right)\) |
| Inequations | \(x_{2} x_{6} x_{10} - x_{2} x_{7} x_{9} - x_{3} x_{5} x_{10} + x_{3} x_{7} x_{8} + x_{4} x_{5} x_{9} - x_{4} x_{6} x_{8} \neq 0\), \(x_{3} x_{7} - x_{3} x_{10} - x_{4} x_{6} + x_{4} x_{9} + x_{6} x_{10} - x_{7} x_{9} \neq 0\), \(x_{2} x_{7} - x_{2} x_{10} - x_{4} x_{5} + x_{4} x_{8} + x_{5} x_{10} - x_{7} x_{8} \neq 0\), \(x_{2} x_{6} x_{10} - x_{2} x_{7} x_{9} + x_{2} x_{7} - x_{2} x_{10} - x_{3} x_{5} x_{10} + x_{3} x_{7} x_{8} - x_{3} x_{7} + x_{3} x_{10} + x_{4} x_{5} x_{9} - x_{4} x_{5} - x_{4} x_{6} x_{8} + x_{4} x_{6} + x_{4} x_{8} - x_{4} x_{9} + x_{5} x_{10} - x_{6} x_{10} - x_{7} x_{8} + x_{7} x_{9} \neq 0\), \(x_{2} x_{6} - x_{2} x_{9} - x_{3} x_{5} + x_{3} x_{8} + x_{5} x_{9} - x_{6} x_{8} \neq 0\), \(x_{1} x_{3} x_{7} - x_{1} x_{3} x_{10} - x_{1} x_{4} x_{6} + x_{1} x_{4} x_{9} + x_{1} x_{6} x_{10} - x_{1} x_{7} x_{9} - x_{2} x_{6} x_{10} + x_{2} x_{6} + x_{2} x_{7} x_{9} - x_{2} x_{9} + x_{3} x_{5} x_{10} - x_{3} x_{5} - x_{3} x_{7} x_{8} + x_{3} x_{8} - x_{4} x_{5} x_{9} + x_{4} x_{6} x_{8} + x_{5} x_{9} - x_{6} x_{8} \neq 0\), \(x_{1} x_{6} x_{10} - x_{1} x_{7} x_{9} + x_{5} x_{9} - x_{6} x_{8} \neq 0\), \(x_{6} x_{10} - x_{6} - x_{7} x_{9} + x_{9} \neq 0\), \(x_{1} x_{7} - x_{1} x_{10} + x_{5} x_{10} - x_{5} - x_{7} x_{8} + x_{8} \neq 0\), \(x_{1} x_{6} x_{10} - x_{1} x_{7} x_{9} + x_{1} x_{7} - x_{1} x_{10} + x_{5} x_{9} + x_{5} x_{10} - x_{5} - x_{6} x_{8} - x_{6} x_{10} + x_{6} - x_{7} x_{8} + x_{7} x_{9} + x_{8} - x_{9} \neq 0\), \(x_{1} x_{6} - x_{1} x_{9} + x_{5} x_{9} - x_{6} x_{8} \neq 0\), \(x_{5} x_{9} - x_{6} x_{8} \neq 0\), \(x_{6} - x_{9} \neq 0\), \(x_{5} - x_{8} \neq 0\), \(x_{5} x_{9} - x_{5} - x_{6} x_{8} + x_{6} + x_{8} - x_{9} \neq 0\), \(x_{5} x_{10} - x_{6} x_{10} - x_{7} x_{8} + x_{7} x_{9} \neq 0\), \(x_{6} x_{10} - x_{7} x_{9} + x_{7} - x_{10} \neq 0\), \(x_{5} x_{10} - x_{7} x_{8} + x_{7} - x_{10} \neq 0\), \(x_{5} x_{10} - x_{7} x_{8} \neq 0\), \(x_{7} - x_{10} \neq 0\), \(x_{6} x_{10} - x_{7} x_{9} \neq 0\), \(x_{1} x_{3} x_{10} - x_{1} x_{4} x_{9} + x_{2} x_{9} - x_{3} x_{8} \neq 0\), \(x_{3} x_{10} - x_{3} - x_{4} x_{9} + x_{9} \neq 0\), \(x_{1} x_{4} - x_{1} x_{10} + x_{2} x_{10} - x_{2} - x_{4} x_{8} + x_{8} \neq 0\), \(x_{1} x_{3} x_{10} - x_{1} x_{4} x_{9} + x_{1} x_{4} - x_{1} x_{10} + x_{2} x_{9} + x_{2} x_{10} - x_{2} - x_{3} x_{8} - x_{3} x_{10} + x_{3} - x_{4} x_{8} + x_{4} x_{9} + x_{8} - x_{9} \neq 0\), \(x_{1} x_{3} - x_{1} x_{9} + x_{2} x_{9} - x_{3} x_{8} \neq 0\), \(x_{2} x_{9} - x_{3} x_{8} \neq 0\), \(x_{3} - x_{9} \neq 0\), \(x_{2} - x_{8} \neq 0\), \(x_{2} x_{9} - x_{2} - x_{3} x_{8} + x_{3} + x_{8} - x_{9} \neq 0\), \(x_{2} x_{10} - x_{3} x_{10} - x_{4} x_{8} + x_{4} x_{9} \neq 0\), \(x_{3} x_{10} - x_{4} x_{9} + x_{4} - x_{10} \neq 0\), \(x_{2} x_{10} - x_{4} x_{8} + x_{4} - x_{10} \neq 0\), \(x_{2} x_{10} - x_{4} x_{8} \neq 0\), \(x_{4} - x_{10} \neq 0\), \(x_{3} x_{10} - x_{4} x_{9} \neq 0\), \(x_{1} \neq 0\), \(x_{9} \neq 0\), \(x_{1} - x_{8} \neq 0\), \(x_{1} x_{9} - x_{1} + x_{8} - x_{9} \neq 0\), \(x_{1} x_{10} - x_{8} + x_{9} \neq 0\), \(x_{9} + x_{10} - 1 \neq 0\), \(x_{1} x_{10} - x_{8} - x_{10} + 1 \neq 0\), \(x_{1} x_{10} - x_{8} \neq 0\), \(x_{10} - 1 \neq 0\), \(x_{8} - x_{9} \neq 0\), \(x_{9} - 1 \neq 0\), \(x_{8} - 1 \neq 0\), \(x_{8} \neq 0\), \(x_{10} \neq 0\), \(x_{1} x_{3} x_{7} - x_{1} x_{4} x_{6} + x_{2} x_{6} - x_{3} x_{5} \neq 0\), \(x_{3} x_{7} - x_{3} - x_{4} x_{6} + x_{6} \neq 0\), \(x_{1} x_{4} - x_{1} x_{7} + x_{2} x_{7} - x_{2} - x_{4} x_{5} + x_{5} \neq 0\), \(x_{1} x_{3} x_{7} - x_{1} x_{4} x_{6} + x_{1} x_{4} - x_{1} x_{7} + x_{2} x_{6} + x_{2} x_{7} - x_{2} - x_{3} x_{5} - x_{3} x_{7} + x_{3} - x_{4} x_{5} + x_{4} x_{6} + x_{5} - x_{6} \neq 0\), \(x_{1} x_{3} - x_{1} x_{6} + x_{2} x_{6} - x_{3} x_{5} \neq 0\), \(x_{2} x_{6} - x_{3} x_{5} \neq 0\), \(x_{3} - x_{6} \neq 0\), \(x_{2} - x_{5} \neq 0\), \(x_{2} x_{6} - x_{2} - x_{3} x_{5} + x_{3} + x_{5} - x_{6} \neq 0\), \(x_{2} x_{7} - x_{3} x_{7} - x_{4} x_{5} + x_{4} x_{6} \neq 0\), \(x_{3} x_{7} - x_{4} x_{6} + x_{4} - x_{7} \neq 0\), \(x_{2} x_{7} - x_{4} x_{5} + x_{4} - x_{7} \neq 0\), \(x_{2} x_{7} - x_{4} x_{5} \neq 0\), \(x_{4} - x_{7} \neq 0\), \(x_{3} x_{7} - x_{4} x_{6} \neq 0\), \(x_{6} \neq 0\), \(x_{1} - x_{5} \neq 0\), \(x_{1} x_{6} - x_{1} + x_{5} - x_{6} \neq 0\), \(x_{1} x_{7} - x_{5} + x_{6} \neq 0\), \(x_{6} + x_{7} - 1 \neq 0\), \(x_{1} x_{7} - x_{5} - x_{7} + 1 \neq 0\), \(x_{1} x_{7} - x_{5} \neq 0\), \(x_{7} - 1 \neq 0\), \(x_{5} - x_{6} \neq 0\), \(x_{6} - 1 \neq 0\), \(x_{5} - 1 \neq 0\), \(x_{5} \neq 0\), \(x_{7} \neq 0\), \(x_{3} \neq 0\), \(x_{1} - x_{2} \neq 0\), \(x_{1} x_{3} - x_{1} + x_{2} - x_{3} \neq 0\), \(x_{1} x_{4} - x_{2} + x_{3} \neq 0\), \(x_{3} + x_{4} - 1 \neq 0\), \(x_{1} x_{4} - x_{2} - x_{4} + 1 \neq 0\), \(x_{1} x_{4} - x_{2} \neq 0\), \(x_{4} - 1 \neq 0\), \(x_{2} - x_{3} \neq 0\), \(x_{3} - 1 \neq 0\), \(x_{2} - 1 \neq 0\), \(x_{2} \neq 0\), \(x_{4} \neq 0\), \(x_{1} - 1 \neq 0\) |
| Realization matrix | \(\begin{pmatrix}1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & x_{1} & x_{2} & x_{5} & x_{8} \\ 0 & 0 & 1 & 1 & 0 & 0 & x_{3} & x_{6} & x_{9} \\ 0 & 0 & 0 & 0 & 1 & 1 & x_{4} & x_{7} & x_{10}\end{pmatrix}\) |
Combinatorics
Computed on the fly from the id.
Bases (124)
{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} {1,3,5,6} {2,3,5,6} {1,4,5,6} {2,4,5,6} {3,4,5,6} {1,2,3,7} {1,2,4,7} {1,3,4,7} {2,3,4,7} {1,2,5,7} {1,3,5,7} {2,3,5,7} {1,4,5,7} {2,4,5,7} {3,4,5,7} {1,2,6,7} {1,3,6,7} {2,3,6,7} {1,4,6,7} {2,4,6,7} {3,4,6,7} {1,5,6,7} {2,5,6,7} {3,5,6,7} {4,5,6,7} {1,2,3,8} {1,2,4,8} {1,3,4,8} {2,3,4,8} {1,2,5,8} {1,3,5,8} {2,3,5,8} {1,4,5,8} {2,4,5,8} {3,4,5,8} {1,2,6,8} {1,3,6,8} {2,3,6,8} {1,4,6,8} {2,4,6,8} {3,4,6,8} {1,5,6,8} {2,5,6,8} {3,5,6,8} {4,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {1,4,7,8} {2,4,7,8} {3,4,7,8} {1,5,7,8} {2,5,7,8} {3,5,7,8} {4,5,7,8} {1,6,7,8} {2,6,7,8} {3,6,7,8} {4,6,7,8} {5,6,7,8} {1,2,3,9} {1,2,4,9} {1,3,4,9} {2,3,4,9} {1,2,5,9} {1,3,5,9} {2,3,5,9} {1,4,5,9} {2,4,5,9} {3,4,5,9} {1,2,6,9} {1,3,6,9} {2,3,6,9} {1,4,6,9} {2,4,6,9} {3,4,6,9} {1,5,6,9} {2,5,6,9} {3,5,6,9} {4,5,6,9} {1,2,7,9} {1,3,7,9} {2,3,7,9} {1,4,7,9} {2,4,7,9} {3,4,7,9} {1,5,7,9} {2,5,7,9} {3,5,7,9} {4,5,7,9} {1,6,7,9} {2,6,7,9} {3,6,7,9} {4,6,7,9} {5,6,7,9} {1,2,8,9} {1,3,8,9} {2,3,8,9} {1,4,8,9} {2,4,8,9} {3,4,8,9} {1,5,8,9} {2,5,8,9} {3,5,8,9} {4,5,8,9} {1,6,8,9} {2,6,8,9} {3,6,8,9} {4,6,8,9} {5,6,8,9} {1,7,8,9} {2,7,8,9} {3,7,8,9} {4,7,8,9} {5,7,8,9} {6,7,8,9}Non-bases (2)
{1,2,3,4} {1,2,5,6}Circuits (118)
{1,2,3,4} {1,2,5,6} {1,3,4,5,6} {2,3,4,5,6} {1,2,3,5,7} {1,2,4,5,7} {1,3,4,5,7} {2,3,4,5,7} {1,2,3,6,7} {1,2,4,6,7} {1,3,4,6,7} {2,3,4,6,7} {1,3,5,6,7} {2,3,5,6,7} {1,4,5,6,7} {2,4,5,6,7} {3,4,5,6,7} {1,2,3,5,8} {1,2,4,5,8} {1,3,4,5,8} {2,3,4,5,8} {1,2,3,6,8} {1,2,4,6,8} {1,3,4,6,8} {2,3,4,6,8} {1,3,5,6,8} {2,3,5,6,8} {1,4,5,6,8} {2,4,5,6,8} {3,4,5,6,8} {1,2,3,7,8} {1,2,4,7,8} {1,3,4,7,8} {2,3,4,7,8} {1,2,5,7,8} {1,3,5,7,8} {2,3,5,7,8} {1,4,5,7,8} {2,4,5,7,8} {3,4,5,7,8} {1,2,6,7,8} {1,3,6,7,8} {2,3,6,7,8} {1,4,6,7,8} {2,4,6,7,8} {3,4,6,7,8} {1,5,6,7,8} {2,5,6,7,8} {3,5,6,7,8} {4,5,6,7,8} {1,2,3,5,9} {1,2,4,5,9} {1,3,4,5,9} {2,3,4,5,9} {1,2,3,6,9} {1,2,4,6,9} {1,3,4,6,9} {2,3,4,6,9} {1,3,5,6,9} {2,3,5,6,9} {1,4,5,6,9} {2,4,5,6,9} {3,4,5,6,9} {1,2,3,7,9} {1,2,4,7,9} {1,3,4,7,9} {2,3,4,7,9} {1,2,5,7,9} {1,3,5,7,9} {2,3,5,7,9} {1,4,5,7,9} {2,4,5,7,9} {3,4,5,7,9} {1,2,6,7,9} {1,3,6,7,9} {2,3,6,7,9} {1,4,6,7,9} {2,4,6,7,9} {3,4,6,7,9} {1,5,6,7,9} {2,5,6,7,9} {3,5,6,7,9} {4,5,6,7,9} {1,2,3,8,9} {1,2,4,8,9} {1,3,4,8,9} {2,3,4,8,9} {1,2,5,8,9} {1,3,5,8,9} {2,3,5,8,9} {1,4,5,8,9} {2,4,5,8,9} {3,4,5,8,9} {1,2,6,8,9} {1,3,6,8,9} {2,3,6,8,9} {1,4,6,8,9} {2,4,6,8,9} {3,4,6,8,9} {1,5,6,8,9} {2,5,6,8,9} {3,5,6,8,9} {4,5,6,8,9} {1,2,7,8,9} {1,3,7,8,9} {2,3,7,8,9} {1,4,7,8,9} {2,4,7,8,9} {3,4,7,8,9} {1,5,7,8,9} {2,5,7,8,9} {3,5,7,8,9} {4,5,7,8,9} {1,6,7,8,9} {2,6,7,8,9} {3,6,7,8,9} {4,6,7,8,9} {5,6,7,8,9}Flats by rank (125)
- rank 0 (1): {}
- rank 1 (9): {1} {2} {3} {4} {5} {6} {7} {8} {9}
- rank 2 (36): {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} {1,7} {2,7} {3,7} {4,7} {5,7} {6,7} {1,8} {2,8} {3,8} {4,8} {5,8} {6,8} {7,8} {1,9} {2,9} {3,9} {4,9} {5,9} {6,9} {7,9} {8,9}
- rank 3 (78): {1,2,3,4} {1,3,5} {2,3,5} {1,4,5} {2,4,5} {3,4,5} {1,3,6} {2,3,6} {1,4,6} {2,4,6} {3,4,6} {1,2,5,6} {3,5,6} {4,5,6} {1,2,7} {1,3,7} {2,3,7} {1,4,7} {2,4,7} {3,4,7} {1,5,7} {2,5,7} {3,5,7} {4,5,7} {1,6,7} {2,6,7} {3,6,7} {4,6,7} {5,6,7} {1,2,8} {1,3,8} {2,3,8} {1,4,8} {2,4,8} {3,4,8} {1,5,8} {2,5,8} {3,5,8} {4,5,8} {1,6,8} {2,6,8} {3,6,8} {4,6,8} {5,6,8} {1,7,8} {2,7,8} {3,7,8} {4,7,8} {5,7,8} {6,7,8} {1,2,9} {1,3,9} {2,3,9} {1,4,9} {2,4,9} {3,4,9} {1,5,9} {2,5,9} {3,5,9} {4,5,9} {1,6,9} {2,6,9} {3,6,9} {4,6,9} {5,6,9} {1,7,9} {2,7,9} {3,7,9} {4,7,9} {5,7,9} {6,7,9} {1,8,9} {2,8,9} {3,8,9} {4,8,9} {5,8,9} {6,8,9} {7,8,9}
- rank 4 (1): {1,2,3,4,5,6,7,8,9}
Hyperplanes (78)
{1,2,3,4} {1,3,5} {2,3,5} {1,4,5} {2,4,5} {3,4,5} {1,3,6} {2,3,6} {1,4,6} {2,4,6} {3,4,6} {1,2,5,6} {3,5,6} {4,5,6} {1,2,7} {1,3,7} {2,3,7} {1,4,7} {2,4,7} {3,4,7} {1,5,7} {2,5,7} {3,5,7} {4,5,7} {1,6,7} {2,6,7} {3,6,7} {4,6,7} {5,6,7} {1,2,8} {1,3,8} {2,3,8} {1,4,8} {2,4,8} {3,4,8} {1,5,8} {2,5,8} {3,5,8} {4,5,8} {1,6,8} {2,6,8} {3,6,8} {4,6,8} {5,6,8} {1,7,8} {2,7,8} {3,7,8} {4,7,8} {5,7,8} {6,7,8} {1,2,9} {1,3,9} {2,3,9} {1,4,9} {2,4,9} {3,4,9} {1,5,9} {2,5,9} {3,5,9} {4,5,9} {1,6,9} {2,6,9} {3,6,9} {4,6,9} {5,6,9} {1,7,9} {2,7,9} {3,7,9} {4,7,9} {5,7,9} {6,7,9} {1,8,9} {2,8,9} {3,8,9} {4,8,9} {5,8,9} {6,8,9} {7,8,9}Dual (rank 5)
Revlex encoding in this labeling (not canonicalized, so not linked): ********************************************************************************************************************0********0
Bases: {4,6,7,8,9} {3,6,7,8,9} {2,6,7,8,9} {1,6,7,8,9} {4,5,7,8,9} {3,5,7,8,9} {2,5,7,8,9} {1,5,7,8,9} {2,4,7,8,9} {1,4,7,8,9} {2,3,7,8,9} {1,3,7,8,9} {1,2,7,8,9} {4,5,6,8,9} {3,5,6,8,9} {2,5,6,8,9} {1,5,6,8,9} {3,4,6,8,9} {2,4,6,8,9} {1,4,6,8,9} {2,3,6,8,9} {1,3,6,8,9} {1,2,6,8,9} {3,4,5,8,9} {2,4,5,8,9} {1,4,5,8,9} {2,3,5,8,9} {1,3,5,8,9} {1,2,5,8,9} {2,3,4,8,9} {1,3,4,8,9} {1,2,4,8,9} {1,2,3,8,9} {4,5,6,7,9} {3,5,6,7,9} {2,5,6,7,9} {1,5,6,7,9} {3,4,6,7,9} {2,4,6,7,9} {1,4,6,7,9} {2,3,6,7,9} {1,3,6,7,9} {1,2,6,7,9} {3,4,5,7,9} {2,4,5,7,9} {1,4,5,7,9} {2,3,5,7,9} {1,3,5,7,9} {1,2,5,7,9} {2,3,4,7,9} {1,3,4,7,9} {1,2,4,7,9} {1,2,3,7,9} {3,4,5,6,9} {2,4,5,6,9} {1,4,5,6,9} {2,3,5,6,9} {1,3,5,6,9} {1,2,5,6,9} {2,3,4,6,9} {1,3,4,6,9} {1,2,4,6,9} {1,2,3,6,9} {2,3,4,5,9} {1,3,4,5,9} {1,2,4,5,9} {1,2,3,5,9} {1,2,3,4,9} {4,5,6,7,8} {3,5,6,7,8} {2,5,6,7,8} {1,5,6,7,8} {3,4,6,7,8} {2,4,6,7,8} {1,4,6,7,8} {2,3,6,7,8} {1,3,6,7,8} {1,2,6,7,8} {3,4,5,7,8} {2,4,5,7,8} {1,4,5,7,8} {2,3,5,7,8} {1,3,5,7,8} {1,2,5,7,8} {2,3,4,7,8} {1,3,4,7,8} {1,2,4,7,8} {1,2,3,7,8} {3,4,5,6,8} {2,4,5,6,8} {1,4,5,6,8} {2,3,5,6,8} {1,3,5,6,8} {1,2,5,6,8} {2,3,4,6,8} {1,3,4,6,8} {1,2,4,6,8} {1,2,3,6,8} {2,3,4,5,8} {1,3,4,5,8} {1,2,4,5,8} {1,2,3,5,8} {1,2,3,4,8} {3,4,5,6,7} {2,4,5,6,7} {1,4,5,6,7} {2,3,5,6,7} {1,3,5,6,7} {1,2,5,6,7} {2,3,4,6,7} {1,3,4,6,7} {1,2,4,6,7} {1,2,3,6,7} {2,3,4,5,7} {1,3,4,5,7} {1,2,4,5,7} {1,2,3,5,7} {1,2,3,4,7} {2,3,4,5,6} {1,3,4,5,6} {1,2,4,5,6} {1,2,3,5,6} {1,2,3,4,6} {1,2,3,4,5}
Loops: none. Parallel classes of size > 1: none.
Downloads
- Record as JSON
- OSCAR:
matroid_from_revlex_basis_encoding("0********0********************************************************************************************************************", 4, 9)