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.

Source : oai:HAL:hal-00958981v1

Volume: Vol. 5

Published on: January 1, 2002

Submitted on: March 26, 2015

Keywords: Rank-3 matroids,Matroid Theory,Unimodality Conjecture,Rank-2 matroids,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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