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We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concave inequal- ities 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. 

Abstract Describing the equality conditions of theAlexandrov–Fenchel inequality[Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81]order polytopesand employs poset theoretic technology. 

In this survey we discuss the notion of combinatorial interpretation in the context of Algebraic Combinatorics and related areas. We approach the subject from the Computational Complexity perspective. We review many examples, state a workable definition, discuss many open problems, and present recent results on the subject. 

We prove that deciding the vanishing of the character of the symmetric group is C=P-complete. We use this hardness result to prove that the square of the character is not contained in #P, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is PP-complete under many-one reductions, and hence PH-hard under Turing-reductions. 

We prove that deciding the vanishing of the character of the symmetric group is C=P-complete. We use this hardness result to prove that the square of the character is not contained in #P, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is PP-complete under many-one reductions, and hence PH-hard under Turing-reductions.