allMatroids -- returns all n-element matroids of rank r

Synopsis

Usage:

allMatroids n

allMatroids(n, r)
Inputs:
- n, an integer, the size of the ground set
- r, an integer, the target rank
Outputs:
- a list, of matroids on n elements

Description

This method returns a list of matroids on n elements of rank r, for small n (currently, n <= 9). This list is complete for isomorphism types of rank r matroids on n elements, i.e. every matroid on n elements of rank r is isomorphic to a unique matroid in this list.

This function will silently switch inputs so that the rank r is the smaller of the two inputs (i.e. allMatroids(3,6) and allMatroids(6,3) return the same output). If no rank r is provided, then all matroids on n elements are returned.

One can perform many verifications using this method:

i1 : L = allMatroids 5; #L

o2 = 38

i3 : all(L, isWellDefined)

o3 = true

i4 : all(subsets(L, 2), S -> quickIsomorphismTest(S#0, S#1) == "false")

o4 = true

i5 : tally(L/fVector/values)

o5 = Tally{{1, 1} => 5              }
        {1, 2, 1} => 6
        {1, 3, 1} => 4
        {1, 3, 3, 1} => 4
        {1, 4, 1} => 2
        {1, 4, 4, 1} => 3
        {1, 4, 6, 1} => 2
        {1, 4, 6, 4, 1} => 2
        {1, 5, 1} => 1
        {1, 5, 5, 1} => 1
        {1, 5, 6, 1} => 1
        {1, 5, 8, 1} => 1
        {1, 5, 8, 5, 1} => 1
        {1, 5, 10, 1} => 1
        {1, 5, 10, 7, 1} => 1
        {1, 5, 10, 10, 1} => 1
        {1, 5, 10, 10, 5, 1} => 1
        {1} => 1

o5 : Tally

i6 : smallMatroids = flatten apply(6, i -> allMatroids i); -- all matroids on < 6 elements

i7 : #smallMatroids

o7 = 70

Ways to use allMatroids:

allMatroids(ZZ)
allMatroids(ZZ,ZZ)

For the programmer

The object allMatroids is a method function.