2024/07/28 by Chan, Swee Hong, Igor Pak, Pak, Igor · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Complexity (cs.CC) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Limits and Structures in Graph Theory #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2407.19608
openalex publication_date 2024/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Stanley--Yan (SY) inequality gives the ultra-log-concavity for the numbers of bases of a matroid which have given sizes of intersections with k fixed disjoint sets. The inequality was proved by Stanley (1981) for regular matroids, and by Yan (2023) in full generality. In the original paper, Stanley asked for equality conditions of the SY~inequality, and proved total equality conditions for regular matroids in the case k=0. In this paper, we completely resolve Stanley's problem. First, we obtain an explicit description of the equality cases of the SY inequality for k=0, extending Stanley's results to general matroids and removing the ``total equality'' assumption. Second, for k≥ 1, we prove that the equality cases of the SY inequality cannot be described in a sense that they are not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level.