This page contains numerical results of a program I wrote to calculate the number of matroids on a finite set of given rank. The non-isomorphic tables listed below are known already but the numbers for the other tables were unknown. If there is any confusion about what is meant, please refer to my thesis.

The number of matroids up to n=8

Matroids of rank-r on Sn, i.e. m(n,r)


The table below gives at the (n,r)th entry the number of matroids of rank r on a ground set of cardinality n.

r\n012345678
0 111 111111
1 137 15 3163127255
2 1736171813401220891
3 1151712053334421022217
4 131813334428520812
5 1634012 1022217
6 112720891
7 1255
8 1
Total 125166840638077516410607540

m(n,1) = n
m(n,2) = b(n+1)-2^n.
, where b(n) = Bell numbers
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Non-isomorphic matroids of rank-r on Sn, f(n,r)


The table below gives at the (n,r)th entry the number of non-isomorphic matroids of rank r on a ground set of cardinality n.

r\n012345678
0111111111
112345678
213713233758
3141338108325
41523108940
51637325
61758
7 1 8
8 1
Total12481738983061724

f(n,1) = n,
f(n,2) = p(1)+p(2)+...+p(n)-n, where p(n) is the number of partitions of the integer n,
The total is as given in A055545
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Loopless matroids of rank-r on Sn, c(n,r)


The table below gives at the (n,r)th entry the number of loopless matroids of rank r on a ground set of cardinality n.

r\n012 3 4 5 6 7 8
0100 0 0 0 0 0 0
111 1 1 1 1 1 1
21 4 14 31 2028764139
3111106123222172803583
4126642283678274374
5 1 57 3592 991829
6 1 120 19903
7 1 247
8 1
Total11262716521355512910094077

c(n,1) = 1
c(n,2) = b(n)-1, the Bell numbers.

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Loopless non-isomorphic matroids of rank-r on Sn, g(n,r)


The table below gives at the (n,r)th entry the number of loopless non-isomorphic matroids of rank r on a ground set of cardinality n.

r\n012345678
0100000000
111111111
21246101421
31392570217
4141885832
51531288
61651
717
81
Total1124921602081418

The values were originally given in papers by Dragan Acketa (1979 and 1984).
g(n,1) = 1,
g(n,2) = p(n)-1, where p(n) is the number of partitions of the integer n,

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Simple Matroids of rank-r on Sn, i.e. s(n,r)


The table below gives at the (n,r)th entry the number of simple matroids of rank r on a ground set of cardinality n.

r\n 2345678
2 1111111
3 15313528389433038
4 116337187007642631
5 1422570 907647
6 19916865
7 1219
8 1
Total 12749733297609000402
Row r=3 of this table is given by 1, (all terms in A056642)+1
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Non-isomorphic Simple Matroids of rank-r on Sn, i.e. nis(n,r)


The table below gives at the (n,r)th entry the number of non-isomorphic simple matroids of rank r on a ground set of cardinality n.

r\n 2345678
2 1111111
3 12492368
4 131149617
5 1422 217
6 1540
7 16
8 1
Total124926101950
The total is as given in A002773
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