关于拟阵交的超对数凹性的注记
A note on the ultra log-concavity of matroid intersection
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中文总结 AI 辅助
本注记证明$M_2^\natural$-凹函数满足超对数凹性，由此得任意一对拟阵的交具有该性质，并通过反例说明该结果无法推广至三个拟阵的交。
中文摘要 AI 辅助
1971年，Mason猜想：拟阵中固定大小的独立集的数量构成一个超对数凹序列。2020年，Brändén和Huh、Anari、Liu、Gharan与Vinzant分别独立证明了该猜想。近期，这一结果被推广至$M^\natural$-凹函数。在本注记中，我们迈出下一步，证明了该性质对$M_2^\natural$-凹函数成立，这也说明任意一对拟阵的交（其本身可能不是拟阵）具有相同性质。此外，我们通过一个分块拟阵的反例，证明该结果无法进一步推广至三个拟阵的交。
英文摘要
In 1971, Mason conjectured that the numbers of independent sets of fixed size in a matroid constitute an ultra log-concave sequence. In 2020, this conjecture was proven by Brändén and Huh and independently by Anari, Liu, Gharan and Vinzant. Recently, this result was extended to $M^\natural$-concave functions. In this note, we make the next step by proving it for $M_2^\natural$-concave functions. This shows the same property for the intersection of any pair of matroids (which itself may not be a matroid). Furthermore, we show that this can not be further extended to the intersections of three matroids by including a counterexample of partition matroids.
