Log-concave poset inequalities
arXiv:2110.10740
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.
arXiv comment70 pages, 4 figures. In v3 additional references are added and typos found by referees are fixed
References in corpus (12)
- The Random-Cluster Model
- Strict unimodality of q-binomial coefficients
- Correlation bounds for fields and matroids
- Hodge-Riemann relations for Potts model partition functions
- A semi-small decomposition of the Chow ring of a matroid
- Extensions of the Kahn--Saks inequality for posets of width two
- The cross-product conjecture for width two posets
- Matroids and log-concavity
- Lower bounds for contingency tables via Lorentzian polynomials
- Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
- Whitney numbers of arrangements via measure concentration of intrinsic volumes
- Topological Tutte Polynomial