Log-concave poset inequalities. (English) Zbl 07910128
Summary: We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concave 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.
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.
MSC:
| 05A20 | Combinatorial inequalities |
| 06A07 | Combinatorics of partially ordered sets |
| 06A06 | Partial orders, general |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
Software:
OEISReferences:
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| [141] | AUTHORS Swee Hong Chan Department of Mathematics, Rutgers University, New Brunswick, NJ, USA sweehong [dot] chan [at] rutgers [dot] edu https://sites.math.rutgers.edu/ sc2518 JOURNAL OF THE ASSOCIATION FOR MATHEMATICAL RESEARCH, 2(1):53-153, 2024 |
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