Matroids/ExampleMatroids - Maple Help

Overview of the Matroids:-ExampleMatroids Subpackage

 Calling Sequence Fano() Hesse() MacLane() NCubeMatroid(n) NonFano() NonPappus() Pappus() TicTacToe() UniformMatroid(r,n) Vamos()

Parameters

 r - integer n - integer

Description

 • There are several standard constructions of matroids in the literature. We list some below.
 – The Fano matroid: A matroid which is not representable over the real numbers, but is representable over the field with two elements.
 – The Hesse matroid: The matroid underlying a Hesse configuration of nine points.
 – The MacLane matroid: Obtained by deleting any element from the ground set of the Hesse matroid. This matroid is non-orientable.
 – The NCube matroid: The matroid underlying the vertices of an $n$ dimensional cube.
 – The NonFano matroid: The matroid obtained by removing one non-basis from the Fano matroid.
 – The NonPappus matroid: The matroid obtained by removing one non-basis from the Pappus matroid.
 – The Pappus matroid: The matroid on nine points realizing the collinearities of Pappus' theorem.
 – The Tic-Tac-Toe matroid: A matroid on nine points whose dual is non-algebraic. It is unknown if the tic-tac-toe matroid is algebraic.
 – The uniform matroid: A matroid where every $r$ subset of $n$ elements is a basis.
 – The Vamos matroid: The smallest matroid which is not representable over any field.

Examples

 > $\mathrm{with}\left(\mathrm{Matroids}\right):$
 > $\mathrm{with}\left(\mathrm{ExampleMatroids}\right):$

Create a matroid from the ExampleMatroids gallery.

 > $M≔\mathrm{UniformMatroid}\left(3,7\right)$
 ${M}{≔}⟨{thⅇ uniform matroiⅆ of rank 3 on 7 ⅇlⅇmⅇnts}⟩$ (1)
 > $\mathrm{evalb}\left(\mathrm{numelems}\left(\mathrm{Bases}\left(M\right)\right)=\mathrm{binomial}\left(7,3\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{AreIsomorphic}\left(\mathrm{Deletion}\left(\mathrm{Hesse}\left(\right),\left\{1\right\}\right),\mathrm{MacLane}\left(\right)\right)$
 ${\mathrm{true}}$ (3)

References

 James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.