相关论文: h-Vectors of matroids and logarithmic concavity

We show that f-vectors of matroid complexes of realisable matroids are log-concave. This was conjectured by Mason in 1972. Our proof uses the recent result by Huh and Katz who showed that the coefficients of the characteristic polynomial of…

In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We…

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we…

We present a number of lower bounds for the h-vectors of k-CM, broken circuit and independence complexes. These lead to bounds on the coefficients of the characteristic and reliability polynomials of matroids. The main techniques are the…

One of the major open questions in matroid theory asks whether the $h$-vector $(h_0,h_1,\ldots,h_s)$ of the broken circuit complex of a matroid $M$ satisfies the following inequalities: $$ h_0\leq h_1\leq \cdots\leq h_{\lfloor s/2\rfloor}…

Stanley conjectured in 1977 that the $h$-vector of a matroid simplicial complex is a pure $O$-sequence. We give simple constructive proofs that the conjecture is true for matroids of rank less than or equal to 3, and corank 2. We used…

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for…

We give two proofs that the $h$-vector of any paving matroid is a pure O-sequence, thus answering in the affirmative a conjecture made by R. Stanley, for this particular class of matroids. We also investigate the problem of obtaining good…

The h-vector of a matroid M is an important invariant related to the independence complex of M and can also be recovered from an evaluation of its Tutte polynomial. A well-known conjecture of Stanley posits that the h-vector of a matroid is…

We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convex cones, and reproves the Hodge-Riemann…

The Rota--Heron--Welsh conjecture (now a theorem of Adiprasito, Huh, and the author) asserts the log-concavity of the characteristic polynomial of matroids. We give an exposition of the Lorentzian polynomial proof following the work of…

Recently, it was proved by Anari-Oveis Gharan-Vinzant, Anari-Liu-Oveis Gharan-Vinzant and Br\"{a}nd\'{e}n-Huh that, for any matroid $M$, its basis generating polynomial and its independent set generating polynomial are log-concave on the…

We study the behavior of $h$-vectors associated to matroid complexes under weak maps, or inclusions of matroid polytopes. Specifically, we show that the $h$-vector of the order complex of the lattice of flats of a matroid is component-wise…

Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we…

A bimatroid is a matroid-like generalization of the collection of regular minors of a matrix. In this article, we use the theory of Lorentzian polynomials to study the logarithmic concavity of natural sequences associated to bimatroids.…

Recently, several proofs of the Mason--Welsh conjecture for matroids have been found, which asserts the log-concavity of the sequence that counts independent sets of a given size. In this article we use the theory of Lorentzian polynomials,…

We show that the h-vector of a ladder determinantal ring cogenerated by M=[u_1|v_1] is log-concave. Thus we prove an instance of a conjecture of Stanley, resp. Conca and Herzog

We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a…

Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate…

We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the…