Enumerating Matroids of Fixed Rank

@article{Pendavingh2015EnumeratingMO,
title={Enumerating Matroids of Fixed Rank},
author={Rudi Pendavingh and Jorn G. van der Pol},
journal={Electron. J. Comb.},
year={2015},
volume={24},
pages={1}
}
• Published 21 December 2015
• Mathematics
• Electron. J. Comb.
It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that $s(n) \sim m(n)$, where $m(n)$ denotes the number of matroids on a fixed groundset of size $n$, and $s(n)$ the number of sparse paving matroids. In an earlier paper, we showed that $\log s(n) \sim \log m(n)$. The bounds that we used for that result were dominated by matroids of rank $r\approx n/2$. In this paper we consider the relation between the number of sparse paving matroids $s(n,r)$ and the…
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References

SHOWING 1-10 OF 34 REFERENCES

• Mathematics, Computer Science
Electron. J. Comb.
• 2015
It is shown that each matroid on $n$ elements has a faithful description consisting of a stable set of a Johnson graph together with a (by comparison) vanishing amount of other information, and using that stable sets in these Johnson graphs correspond one-to-one to sparse paving matroids on n elements.
• Mathematics
Comb.
• 2018
It is derived that asymptotically almost all matroids on n elements have a Uk,2k-minor, whenever k≤O(log(n), (2) have girth ≥Ω(n%), (3) have Tutte connectivity, and (4) do not arise as the truncation of another matroid.
• Computer Science, Mathematics
Comb.
• 2015
This work proves an upper bound on loglogmn that is within an additive 1+o(1) term of Knuth’s lower bound, based on using some structural properties of non-bases in a matroid together with some properties of stable sets in the Johnson graph to give a compressed representation of matroids.
• Mathematics
Combinatorics, Probability and Computing
• 2019
It is shown that with high probability the proportion of vertices of ${\cal H}$ that are not saturated by the final matching is at most (L/D)(1/(2(r−1)))+o(1), which is a natural barrier in the analysis of the random greedy hypergraph matching process.
• Computer Science, Mathematics
IEEE Trans. Inf. Theory
• 1980
Several lower bounds for A(n,2\delta,w) are given, better than the "Gilbert bound" in most cases.
• S. Jukna
• Mathematics
Texts in Theoretical Computer Science. An EATCS Series
• 2001
This second edition of this introduction to extremal combinatorics for nonspecialists has been extended with substantial new material, and has been revised and updated throughout.
2. Count the number of edge colorings of a square, up to rotation and reflection. Give them all explicitly for m = 2. Now the group has eight elements: id, (1234), (13)(24), (1432), (13), (24),
• Mathematics
• 1970
It has been clear within the last ten years that combinatorial geometry, together with its order-theoretic counterpart, the geometric lattice, can serve to catalyze the whole field of combinatorial