2013
On the number of matroids
Publication
Publication
Presented at the
ACM-SIAM Symposium on Discrete Algorithms
We consider the problem of determining mn, the number of matroids on n elements.
The best known lower bound on mn is due to Knuth (1974) who showed that
log log mn is at least n−
3
2
log n−O(1). On the other hand, Piff (1973) showed that
log log mn ≤ n − log n + log log n + O(1), and it has been conjectured since that the
right answer is perhaps closer to Knuth’s bound.
We show that this is indeed the case, and prove an upper bound on log log mn
that is within an additive 1 + o(1) term of Knuth’s lower bound. Our proof is
based on using some structural properties of non-bases in a matroid together with
some properties of independent sets in the Johnson graph to give a compressed
representation of matroids.
| Additional Metadata | |
|---|---|
| ACM-SIAM Symposium on Discrete Algorithms | |
| Organisation | Computer Security |
|
Bansal, N., Pendavingh, R., & van der Pol, J. (2013). On the number of matroids. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms 2013 (SODA 0) (pp. 451–471). |
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