Strictness of the log-concavity of generating polynomials of matroids

Murai, Shinji; Nagaoka, Takahiro; Yazawa, Akiko

doi:10.1016/j.jcta.2020.105351

Cited by 6 publications

(2 citation statements)

References 5 publications

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“…It seems, this is the case of our equality conditions for matroid log-concavity given in §1.9 (see also §1.13). In the context of this paper, the only nontrivial equality condition known prior to this work for matroid inequalities is Theorem 1.8 proved by Murai, Nagaoka and Yazawa in [MNY21] using an algebraic argument built on [BH20].…”

Section: 12mentioning

confidence: 99%

Log-concave poset inequalities

Chan,

Pak

2024

JAMR

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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.

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Section: 12mentioning

confidence: 99%

Log-concave poset inequalities

Chan,

Pak

2024

JAMR

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“…Curiously, the equality cases for these inequalities are rather trivial and can be verified in polynomial time [MNY21] (see also [CP21,§1.6]). Here we assume that the matroid is given in a concise presentation (such presentations include graphical, bicircular and representable matroids).…”

Section: Fixing One Elementmentioning

confidence: 99%

The extremals of the Alexandrov–Fenchel inequality for convex polytopes

Shenfeld,

van Handel

2023

Acta Mathematica

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Describing the equality conditions of the Alexandrov-Fenchel inequality 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 [SvH20], which gave a geometric characterization of the equality conditions. The proof involves Stanley's order polytopes and employs poset theoretic technology.

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The eigenvalues of the Hessian matrices of the generating functions for trees with k components

Yazawa

2021

Linear Algebra and its Applications

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Strictness of the log-concavity of generating polynomials of matroids

Cited by 6 publications

References 5 publications

Log-concave poset inequalities

Log-concave poset inequalities

The extremals of the Alexandrov–Fenchel inequality for convex polytopes

The eigenvalues of the Hessian matrices of the generating functions for trees with k components

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