Related papers: Internally Perfect Matroids

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…

In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This…

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable…

A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible $h$-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve…

A well-known conjecture of Stanley is that the h-vector of any matroid is a pure O-sequence. There have been numerous papers with partial progress on this conjecture, but it is still wide open. Positroids are special class of linear…

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…

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…

An order ideal is a finite poset X of (monic) monomials such that, whenever M is in X and N divides M, then N is in X. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector,…

It is shown that the h-vectors of Stanley-Reisner rings of three classes of matroids are pure O-sequences. The classes are (a) matroids that are truncations of other matroids, or more generally of Cohen-Macaulay complexes, (b) matroids…

Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a r-dimensional…

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 show that the $h$-vector of a $2$-dimensional PS ear-decomposable simplicial complex is a pure $\mathcal{O}$-sequence. This provides a strengthening of Stanley's conjecture for matroid $h$-vectors in rank $3$. Our approach modifies the…

In $1980$ White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our…

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A…

We study the algebraic matroid induced by the ideal of (r+1)-minors of a matrix of variables over a field. This is inherently connected to the bounded-rank matrix completion problem, in which the aim is to complete a partially observed rank…

It is proved that the broken circuit complex of an ordered matroid is Gorenstein if and only if it is a complete intersection. Several characterizations for a matroid that admits such an order are then given, with particular interest in the…

This paper investigates infinite matroids from a categorical perspective. We prove that the category of infinite matroids is a proto-exact category in the sense of Dyckerhoff and Kapranov, thereby generalizing our previous result on the…

Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the…

We discuss a conjecture of Ingleton on excluded minors for base-orderability, and, extending a result he stated, we prove that infinitely many of the matroids that he identified are excluded minors for base-orderability, as well as for the…

A well-known conjecture of Stanley is that the h-vector of a matroid is a pure O-sequence. There have been numerous papers with partial progress on this conjecture, but it is still wide open. In particular, for graphic matroids coming from…