fVector(Matroid) -- f-vector of a matroid

Synopsis

Function: fVector
Usage:

fVector M
Inputs:
- M, a matroid,
Outputs:
- a hash table,

Description

The f-vector of a matroid M is the invariant (f_0, f_1, ..., f_r), where f_i is the number of rank i flats of M, and r is the rank of M. Note that f_0 = f_r = 1, as the set of loops is the unique flat of rank 0, and the ground set is the unique flat of maximal rank.

i1 : M = matroid({a,b,c,d},{{a,b},{a,c}})

o1 = a "matroid" of rank 2 on 4 elements

o1 : Matroid

i2 : fVector M

o2 = HashTable{0 => 1}
               1 => 2
               2 => 1

o2 : HashTable

i3 : fVector matroid completeGraph 4

o3 = HashTable{0 => 1}
               1 => 6
               2 => 7
               3 => 1

o3 : HashTable

Caveat

This is not the same as the f-vector of the independence complex of the matroid M, which counts the number of independent sets of a given size. To do this instead, use "fVector independenceComplex M".

Ways to use this method: