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On a Unimodality Conjecture in Matroid Theory

Abstract : A certain unimodal conjecture in matroid theory states the number of rank-r matroids on a set of size n is unimodal in r and attains its maximum at r=\lfloor n/2 \rfloor . We show that this conjecture holds up to r=3 by constructing a map from a class of rank-2 matroids into the class of loopless rank-3 matroids. Similar inequalities are proven for the number of non-isomorphic loopless matroids, loopless matroids and matroids.
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W. M. B. Dukes. On a Unimodality Conjecture in Matroid Theory. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2002, 5, pp.181-190. ⟨hal-00958981⟩

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