Log-concave poset inequalities

Abstract: In the ocean of log-concave inequalities, there are two islands that are especially difficult. First, Mason's conjectures say that the number of forests in a graph with k edges is log-concave. More generally, the number of independent sets of size k in a matroid is log-concave. Versions of these results were established just recently, in a remarkable series of papers inspired by algebraic and geometric considerations. Second, Stanley's inequality for the numbers of linear extensions of a poset with value k at a given poset element, is log-concave. This was originally conjectured by Chung, Fishburn and Graham, and proved by Stanley in 1981 using the Alexandrov–Fenchel inequalities in convex geometry. In our recent paper, we present a new framework of combinatorial atlas which allows one to give elementary proofs of both results, and extend them in several directions. I will give an introduction to the area and then outline our approach. Joint work with Swee Hong Chan.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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