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\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | | 1 | 3 | 7 | 15 | 31 | 63 | 127 | 255 |
2 | | | 1 | 7 | 36 | 171 | 813 | 4012 | 20891 |
3 | | | | 1 | 15 | 171 | 2053 | 33442 | 1022217 |
4 | | | | | 1 | 31 | 813 | 33442 | 8520812 |
5 | | | | | | 1 | 63 | 4012 | 1022217 |
6 | | | | | | | 1 | 127 | 20891 |
7 | | | | | | | | 1 | 255 |
8 | | | | | | | | | 1 |
Total | 1 | 2 | 5 | 16 | 68 | 406 | 3807 | 75164 | 10607540 |
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\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
2 | | | 1 | 3 | 7 | 13 | 23 | 37 | 58 |
3 | | | | 1 | 4 | 13 | 38 | 108 | 325 |
4 | | | | | 1 | 5 | 23 | 108 | 940 |
5 | | | | | | 1 | 6 | 37 | 325 |
6 | | | | | | | 1 | 7 | 58 |
7 | | | | | | | | 1 | 8 |
8 | | | | | | | | | 1 |
Total | 1 | 2 | 4 | 8 | 17 | 38 | 98 | 306 | 1724 |
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\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | | | 1 | 4 | 14 | 31 | 202 | 876 | 4139 |
3 | | | | 1 | 11 | 106 | 1232 | 22172 | 803583 |
4 | | | | | 1 | 26 | 642 | 28367 | 8274374 |
5 | | | | | | 1 | 57 | 3592 | 991829 |
6 | | | | | | | 1 | 120 | 19903 |
7 | | | | | | | | 1 | 247 |
8 | | | | | | | | | 1 |
Total | 1 | 1 | 2 | 6 | 27 | 165 | 2135 | 55129 | 10094077 |
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\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | | | 1 | 2 | 4 | 6 | 10 | 14 | 21 |
3 | | | | 1 | 3 | 9 | 25 | 70 | 217 |
4 | | | | | 1 | 4 | 18 | 85 | 832 |
5 | | | | | | 1 | 5 | 31 | 288 |
6 | | | | | | | 1 | 6 | 51 |
7 | | | | | | | | 1 | 7 |
8 | | | | | | | | | 1 |
Total | 1 | 1 | 2 | 4 | 9 | 21 | 60 | 208 | 1418 |
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 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | | 1 | 5 | 31 | 352 | 8389 | 433038 |
4 | | | 1 | 16 | 337 | 18700 | 7642631 |
5 | | | | 1 | 42 | 2570 | 907647 |
6 | | | | | 1 | 99 | 16865 |
7 | | | | | | 1 | 219 |
8 | | | | | | | 1 |
Total | 1 | 2 | 7 | 49 | 733 | 29760 | 9000402 |
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 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | | 1 | 2 | 4 | 9 | 23 | 68 |
4 | | | 1 | 3 | 11 | 49 | 617 |
5 | | | | 1 | 4 | 22 | 217 |
6 | | | | | 1 | 5 | 40 |
7 | | | | | | 1 | 6 |
8 | | | | | | | 1 |
Total | 1 | 2 | 4 | 9 | 26 | 101 | 950 |
The total is as given in A002773
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