MatroidTools
Matroids from bases: rank, duality, deletion/contraction, Tutte polynomials, transversal, lattice path and rook matroids.
Load with Needs["MatroidTools`"].
Background on symmetricfunctions.com: Matroids, Lattice path matroids.
Functions and symbols
Basis01Vector
Basis01Vector[bases] or Basis01Vector[groundSet,basis] returns a 01-vector representing the basis, or a list of such.
Background: Matroids
ExternallyActiveElements
ExternallyActiveElements[groundSet,bases,basis] returns the externally active elements of basis with respect to the order on groundSet.
Background: Matroids
IndependentSets
IndependentSets[bases] returns all independent sets contained in the listed bases.
Background: Matroids
InternallyActiveElements
InternallyActiveElements[groundSet,bases,basis] returns the internally active elements of basis with respect to the order on groundSet.
Background: Matroids
IsMatroidQ
Given a list of sets, see if they satisfy the basis exchange axioms.
Background: Matroids
IsSubsetClosedQ
IsSubsetClosedQ[sets] returns true if every subset of every set in list of sets is also in the list of sets.
Background: Matroids
LatticeOfFlats
LatticeOfFlats[bases] returns all ordered comparable pairs of distinct flats, represented as pairs of sets.
Background: Matroids
MatchingMatroidBases
MatchingMatroidBases[g] returns the bases for the matching matroid associated with g.
Background: Matroids
MatroidCharacteristicPolynomial
MatroidCharacteristicPolynomial[{groundSet,bases},t] returns the characteristic polynomial.
Background: Matroids
MatroidColoops
MatroidColoops[groundSet,bases] returns the list of coloops, namely elements contained in every basis.
Background: Matroids
MatroidContraction
MatroidContraction[bases,e] contracts with respect to element(s) e in the matroid.
Background: Matroids
MatroidDeletion
MatroidDeletion[bases,e] deletes the element(s) e from the matroid.
Background: Matroids
MatroidDual
MatroidDual[groundSet,bases] returns the bases of the dual matroid.
Background: Matroids
MatroidFlats
MatroidFlats[bases] returns the flats of the matroid specified by its bases.
Background: Matroids
MatroidIsomorphisms
MatroidIsomorphisms[basesA,basesB] returns all bijections carrying the bases of one matroid to the bases of the other.
Background: Matroids
MatroidLoops
MatroidLoops[groundSet,bases] returns the list of loops (dependent 1-element sets).
Background: Matroids
MatroidSetRank
MatroidSetRank[bases,set] returns the matroid rank of set, computed as the maximum intersection size with a basis.
Background: Matroids
MatroidTuttePolynomial
MatroidTuttePolynomial[{groundSet,bases},{x,y}] returns the Tutte polynomial associated with the matroid.
Background: Matroids
MConvexSetQ
MConvexSetQ[sets] returns true if these sets form an M-convex set (of sets).
Background: Matroids
PathBases
PathBases[lam,mu] returns the path matroid bases associated with the skew shape.
Background: Matroids
PathSetSystem
PathSetSystem[lam,mu] returns the set system that gives rise to LPMs.
Background: Matroids
RookBases
RookBases[lam,mu] returns the rook matroid bases associated with the skew shape.
Background: Matroids
RookSetSystem
RookSetSystem[lam,mu] returns a set system to produce the rook matroid.
Background: Matroids
SetIsomorphisms
SetIsomorphisms[setsA,setsB] returns all isomorphisms between the two sets.
Background: Matroids
SetsGeneratingPolynomial
SetsGeneratingPolynomial[sets,x] returns the multivariate set generating polynomial. Each set {a1,a2,…,ak} contributes with a monomial x[a1]x[a2]…x[ak].
Background: Matroids
SetSymmetries
SetSymmetries[sets] returns a list of lists,
each list describes a permutation group which is in the automorphism group the collection of sets.
Background: Matroids
TransversalBases
TransversalBases[sets] returns all complete transversals of the list of sets.
Background: Matroids
UniformBases
UniformBases[r,n] returns all r-element bases of the uniform matroid on {1,2,…,n}.
Background: Matroids
VamosBases
VamosBases[] returns the list of bases of the non-realizable Vamos matroid.
Background: Matroids