Log-Concavity of Flats of Matroids
Statement
Mason conjectured the following: let be a matroid of rank , and let denote the number of flats of of rank . Is it true that for all , we have ? This is false; a counterexample is given by a graphic matroid whose graph is a generalized theta graph with edges.
Context
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
The flats version of Mason's 1972 conjecture, and the less cited of the two statements that carry his name - the famous one concerns independent sets. Placed below Rota's flat unimodality conjecture at 36, which asks the weaker question about the same sequence and was refuted three weeks later.
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The author prompted ChatGPT-5.5 Pro to search for counterexamples to Mason's conjecture. After finding none on at most elements, the author expanded his search to consider matroids on large ground sets realizable over and at failures of log-concavity at high indices. ChatGPT-5.5 Pro found a variant of the given counterexample.
ChatGPT-5.5 Pro and Claude Opus 4.8 were used for proofreading and generating the figures.
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