AI 中文总结
该研究反驳了罗塔关于拟阵平集计数为单峰序列的猜想，通过广义θ图、直和及Whittle的q提升构造，证明拟阵平集计数可存在任意多个峰值。
AI 中文摘要
我们反驳了罗塔（Rota）的猜想，即拟阵中按秩划分的平集计数构成单峰序列。此外，我们证明该序列可以有任意多个峰值。构造始于找到一个严重违反对数凹性的广义θ图，通过直和在多处破坏对数凹性，再使用惠特尔（Whittle）的q提升构造生成一个平集计数具有多个峰值的拟阵。
英文摘要
We disprove Rota's conjecture that the counts of flats in a matroid according to rank form a unimodal sequence. Furthermore, we show that this sequence can have arbitrarily many peaks. The construction starts by finding a generalized theta graph for which log-concavity fails severely. By taking direct sums, we break log-concavity in many places. We then use Whittle's $q$-lift construction to produce a matroid whose flat counts have many peaks.
Comments9 pages, 1 figure; supersedes arXiv:2607.22515 and section 2 of arXiv:2607.02208
