  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.

Combinatorics;Discrete Mathematics