A note on the ultra log-concavity of matroid intersection

Adam Schweitzer

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arXiv · 2608.23262

A note on the ultra log-concavity of matroid intersection

Adam Schweitzer

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Abstract

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 Br\"and\'en 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.

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Adam Schweitzer. 2026-08-24. A note on the ultra log-concavity of matroid intersection. https://arxiv.org/abs/2608.23262

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