Log-concave poset inequalities

TL;DR

This work develops a unified, elementary but powerful framework for log-concavity inequalities across a spectrum of combinatorial structures: matroids, discrete polymatroids, poset antimatroids, and interval greedoids. Central to the approach is the combinatorial atlas, an acyclic graph whose vertices carry nonnegative symmetric matrices and vectors, with edge maps guiding recursive reductions; hyperbolicity of these matrices yields the desired LC inequalities for weighted counts of feasible words, independent sets, and linear extensions. The paper not only proves broad LC inequalities in this framework but also gives precise equality conditions, often revealing that equality occurs only in highly structured cases (e.g., uniform weights, girth constraints, tree-poset configurations, or specific Steiner-system/finite-field examples). A key feature is the chain of reductions that transports LC and equality results from interval greedoids to poset antimatroids, polymatroids, and matroids, while also re-deriving and extending Stanley-type inequalities for linear extensions with weighted weights. The methods provide new tools for understanding equality phenomena and open avenues for applying hyperbolic-matrix atlases to other greedoid-like combinatorial families.

Abstract

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.

Figures (3)

• Figure 8.1: Edges of two type: $\mathop{\mathrm{\text{\it e}}}\nolimits^{\langle x\rangle} = \bigl(\mathop{\mathrm{\text{\it v}}}\nolimits,\mathop{\mathrm{\text{\it v}}}\nolimits^{\langle x\rangle}\bigr)$, $v=(\alpha,m,t)$, $\mathop{\mathrm{\text{\it v}}}\nolimits^{\langle x\rangle}=(\alpha x,m-1,1)$, and $\mathop{\mathrm{\text{\it e}}}\nolimits^{\langle\textsf{null}\rangle} = \bigl(\mathop{\mathrm{\text{\it v}}}\nolimits,\mathop{\mathrm{\text{\it v}}}\nolimits^{\langle\textsf{null}\rangle}\bigr)$, $v=(\alpha,m,t)$, $\mathop{\mathrm{\text{\it v}}}\nolimits^{\langle\textsf{null}\rangle}=(\alpha,m-1,1)$.
• Figure 17.1: Diagram of inclusions of greedoid classes.
• Figure :

Theorems & Definitions (137)

• Theorem 1.1: Log-concavity for matroids, AHK, formerly Welsh--Mason conjecture
• Theorem 1.2: One-sided ultra-log-concavity for matroids, HSW, formerly weak Mason conjecture
• Theorem 1.3: Ultra-log-concavity for matroids, ALOV and BH, formerly strong Mason conjecture
• Theorem 1.4: Refined log-concavity for matroids
• Example 1.5: Graphical matroids
• Theorem 1.6: Refined weighted log-concavity for matroids
• Remark 1.7
• Theorem 1.8: Equality for matroids, MNY
• Theorem 1.9: Weighted equality for matroids
• Theorem 1.10: Refined equality for matroids