Matroid 4.9.189749

Label	4.9.189749
Id	`r4_n9_00000000000000000000000000000000000000000000000000000000000000000000000000000*000000000000000*00`
Rank	4
n	9

A graph with this cycle matroid; edge labels are elements.

Basic invariants

Bases	32
Circuits	11
Flats	35
Cyclic flats	9
Loops	1
Connected components	2
Automorphisms	16
Beta invariant	0
Girth	1
Simple	no
Connected	no
Uniform	not computed
Loopless	no
Coloopless	yes
Paving	no
Laminar	no
Nested	no
Self dual	no
Identically self dual	no
Series parallel	no
Transversal	yes
Supersolvable	yes
Divisionally free	yes
Orientable	yes
Three lines	not computed

Representability

Characteristic set	characteristic 0 and all primes `[0]`
Realizable	yes
Realizable char0	yes
Regular	yes
Binary	yes
Ternary	yes
Quaternary	yes
Graphic	yes

Realization space

Status	computed_realization_space
Dimension of realization space	3
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: `rigid`
Birational type components	show [ { "dim": 0, "free_rank": 0, "torus_rank": 0, "core_dim": 0, "qbar_components": 1, "type": "rational", "reason": "rigid: unique realization over Q (integer matrix)" } ]
Good basis	not computed
Good basis v2	2,4,6,8

Geometry

Realization space: scheme simple core id	not computed
Smooth char 0	yes
Singular primes	[]
Realization space: smooth over ℚ	not computed
Realization space: smooth over ℤ	not computed
Realization space: is regular scheme	not computed
Realization space: singular fiber primes	not computed
Realization space: singular fiber characteristics	not computed
Realization space: characteristic dimensions	not computed
Realization space: characteristic dimension unexpected	not computed
Realization space: characteristic dimension varies	not computed

Other

Char poly splits	true
Simplicial arrangement	no
Realization space: n qbar components	not computed

Tutte polynomial

\(T = x_{0}^{4} x_{1} + x_{0}^{3} x_{1}^{2} + 3 x_{0}^{3} x_{1} + x_{0}^{2} x_{1}^{3} + 5 x_{0}^{2} x_{1}^{2} + 3 x_{0}^{2} x_{1} + 2 x_{0} x_{1}^{4} + 5 x_{0} x_{1}^{3} + 4 x_{0} x_{1}^{2} + x_{0} x_{1} + x_{1}^{5} + 2 x_{1}^{4} + 2 x_{1}^{3} + x_{1}^{2}\)

Realization space

Ring	\(\mathbb{Z}\)
Defining ideal	\(\left(0\right)\)
Inequations	none
Realization matrix	\(\begin{pmatrix}0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}\)

Combinatorics

Computed on the fly from the id.

Bases (32)

{2,4,6,8} {3,4,6,8} {2,5,6,8} {3,5,6,8} {4,5,6,8} {2,4,7,8} {3,4,7,8} {2,5,7,8} {3,5,7,8} {4,5,7,8} {2,6,7,8} {3,6,7,8} {5,6,7,8} {2,4,6,9} {3,4,6,9} {2,5,6,9} {3,5,6,9} {4,5,6,9} {2,4,7,9} {3,4,7,9} {2,5,7,9} {3,5,7,9} {4,5,7,9} {2,6,7,9} {3,6,7,9} {5,6,7,9} {2,6,8,9} {3,6,8,9} {5,6,8,9} {2,7,8,9} {3,7,8,9} {5,7,8,9}

Non-bases (94)

{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} {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} {1,5,6,8} {1,2,7,8} {1,3,7,8} {2,3,7,8} {1,4,7,8} {1,5,7,8} {1,6,7,8} {4,6,7,8} {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} {1,5,6,9} {1,2,7,9} {1,3,7,9} {2,3,7,9} {1,4,7,9} {1,5,7,9} {1,6,7,9} {4,6,7,9} {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} {4,6,8,9} {1,7,8,9} {4,7,8,9} {6,7,8,9}

Circuits (11)

{1} {2,3} {2,4,5} {3,4,5} {4,6,7} {2,5,6,7} {3,5,6,7} {4,8,9} {2,5,8,9} {3,5,8,9} {6,7,8,9}

Flats by rank (35)

rank 0 (1): {1}
rank 1 (7): {1,2,3} {1,4} {1,5} {1,6} {1,7} {1,8} {1,9}
rank 2 (15): {1,2,3,4,5} {1,2,3,6} {1,5,6} {1,2,3,7} {1,5,7} {1,4,6,7} {1,2,3,8} {1,5,8} {1,6,8} {1,7,8} {1,2,3,9} {1,5,9} {1,6,9} {1,7,9} {1,4,8,9}
rank 3 (11): {1,2,3,4,5,6,7} {1,2,3,6,8} {1,5,6,8} {1,2,3,7,8} {1,5,7,8} {1,2,3,6,9} {1,5,6,9} {1,2,3,7,9} {1,5,7,9} {1,2,3,4,5,8,9} {1,4,6,7,8,9}
rank 4 (1): {1,2,3,4,5,6,7,8,9}

Hyperplanes (11)

{1,2,3,4,5,6,7} {1,2,3,6,8} {1,5,6,8} {1,2,3,7,8} {1,5,7,8} {1,2,3,6,9} {1,5,6,9} {1,2,3,7,9} {1,5,7,9} {1,2,3,4,5,8,9} {1,4,6,7,8,9}

Dual (rank 5)

Revlex encoding in this labeling (not canonicalized, so not linked): 0*0**0*0**00000000000*0**0***0**0000***0**00000000000000*0**0***0**0000***0**0000000000000000000000000000000000000000000000000

Bases: {1,3,5,7,9} {1,2,5,7,9} {1,3,4,7,9} {1,2,4,7,9} {1,2,3,7,9} {1,3,5,6,9} {1,2,5,6,9} {1,3,4,6,9} {1,2,4,6,9} {1,2,3,6,9} {1,3,4,5,9} {1,2,4,5,9} {1,2,3,4,9} {1,3,5,7,8} {1,2,5,7,8} {1,3,4,7,8} {1,2,4,7,8} {1,2,3,7,8} {1,3,5,6,8} {1,2,5,6,8} {1,3,4,6,8} {1,2,4,6,8} {1,2,3,6,8} {1,3,4,5,8} {1,2,4,5,8} {1,2,3,4,8} {1,3,4,5,7} {1,2,4,5,7} {1,2,3,4,7} {1,3,4,5,6} {1,2,4,5,6} {1,2,3,4,6}

Loops: 1. Parallel classes of size > 1: {2,3}.

Downloads

Record as JSON
OSCAR: matroid_from_revlex_basis_encoding("0000000000000000000000000000000000000000000000000**0***0000**0***0**0*00000000000000**0***0000**0***0**0*00000000000**0*0**0*0", 4, 9)