Log-concavity of the sequence of numbers of matroid flats

Prove that, for every matroid M, the sequence c_0,\dots,c_d, where c_k is the number of flats of M of rank k, is log-concave, potentially by applying the paper’s log-concavity theorem for colored path complexes.

Background

The paper develops a log-concavity theorem for sequences derived from colored path complexes and applies it to characteristic-polynomial coefficients and several sequences associated with distributive lattices. It then identifies a distinct conjectural sequence for an arbitrary matroid: the rank-level counts of flats.

The proposed approach is to determine whether the colored-path-complex log-concavity theorem can be adapted to establish log-concavity of these flat-number sequences. The problem is explicitly presented as a conjecture rather than as a consequence proved in the paper.