Small Matroids
In the late 1960s Blackburn, Crapo & Higgs computed a catalogue of
all the matroids on 8 elements which was not extended for more than 35
years - this must surely make it one of the longest standing
computational results in combinatorics. The cover of the original
technical report describing this work is a masterpiece of 60s-style
"album-cover" stylised lettering (click on the image to see if you can read the title).
Dillon Mayhew and I have now extended this catalogue by determining all of the matroids on 9 elements. Although this may seem rather unambitious given that we have the benefit of 30+ years of advances in techniques in computational combinatorics along with 30+ years of increased computational power, we will see that even computing the numbers of matroids on 10 elements is likely to be out of reach for some time to come.
| Size Rank | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 2 | 1 | 3 | 7 | 13 | 23 | 37 | 58 | 87 | ||
| 3 | 1 | 4 | 13 | 38 | 108 | 325 | 1275 | |||
| 4 | 1 | 5 | 23 | 108 | 940 | 190214 | ||||
| 5 | 1 | 6 | 37 | 325 | 190214 | |||||
| 6 | 1 | 7 | 58 | 1275 | ||||||
| 7 | 1 | 8 | 87 | |||||||
| 8 | 1 | 9 | ||||||||
| 9 | 1 | |||||||||
| Totals | 1 | 2 | 4 | 8 | 17 | 38 | 98 | 306 | 1724 | 383172 |
More Information
MatroidSeeker is a web-interface for counting the numbers of matroids with various properties.
MatroidBrowser is a web-interface for browsing individual matroids and navigating through the catalogue.
The Oxley-list identifies matroids found in Oxley's book within the catalogue.
Last updated by Gordon Royle on 14-Feb-07