We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concavity inequalities for counting certain weighted feasible words, which generalize and extend several previous results establishing Mason conjectures for the numbers of independent sets of matroids. Notably, we prove matching equality conditions for both earlier inequalities and our extensions. In contrast with much of the previous work, our proofs are combinatorial and employ nothing but linear algebra. We use the language formulation of greedoids which allows a linear algebraic setup, which in turn can be analyzed recursively. The underlying non-commutative nature of matrices associated with greedoids allows us to proceed beyond polymatroids and prove the equality conditions. As further application of our tools, we rederive both Stanley's inequality on the number of certain linear extensions, and its equality conditions, which we then also extend to the weighted case.

翻译：我们研究各类集合系统（拟阵、多拟阵、偏序反拟阵及区间贪心oids）的组合不等式。通过证明计数加权可行词的log-concave不等式，我们推广并扩展了多个先前关于拟阵独立集数量的Mason猜想结果。值得注意的是，我们在早期不等式及新推广结果中均证明了匹配的等式条件。与以往多数工作不同，我们的证明完全基于线性代数且具有组合学特性。利用贪心oids的语言框架，我们构建线性代数模型，并采用递归方法进行分析。与贪心oids关联矩阵的非交换性质使我们得以超越多拟阵范畴，成功证明等式条件。作为工具的新应用，我们重新推导了Stanley关于特定线性扩展数量的不等式及其等式条件，并将其推广至加权情形。