Rota’s flat-count unimodality conjecture for matroids
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For a matroid, count flats of each rank. Must this sequence be unimodal, increasing to one peak and then decreasing? This is Rota’s flat-count unimodality conjecture.
Why it matters and what counts as progress
Why it matters: Unimodality asks for a basic shape constraint weaker than the already disproved flat-count log-concavity claim.
What counts as progress: A universal unimodality proof or an explicit matroid whose flat counts have a genuine later rise after a fall.
How the problem developed
- 1971 · Conjecture [1]
Gian-Carlo Rota Posed the conjecture that the sequence counting flats by rank in a matroid is unimodal.
- 2 Jul 2026 · Counterexample [2]
Spencer Larson Refuted a stronger log-concavity assertion for matroid flat counts, without by itself producing a non-unimodal sequence.
- 24 Jul 2026 · Counterexample [3]
Authors of the Rota counterexample paper Constructed matroids with non-unimodal flat-count sequences using a q-lift of graphic matroids.
AI contribution
Matroid flat counts are not unimodal
The authors adapt a ChatGPT-supplied q-lift example against reciprocal flat-count convexity into matroids whose numbers of flats by rank are not unimodal. The final construction combines Larson’s graphic matroids with Whittle’s q-lift.
Sources
Website timeline
Publication dates and editorial changes.
Authors report a ChatGPT-seeded counterexample to Rota’s flat-unimodality conjecture