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# 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 .