Related papers: Enumerating Matroids and Linear Spaces

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank $3$): (a) The number of extreme points in an…

The rank of the matrix multiplication operator for nxn matrices is one of the most studied quantities in algebraic complexity theory. I prove that the rank is at least n^2-o(n^2). More precisely, for any integer p\leq n -1, the rank is at…

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to $t$-branch split cuts and $t$-dimensional lattice cuts for a fixed positive integer $t$. These inequalities are the same as split cuts when $t=1$ and…

This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…

A simple binary matroid is called claw-free if none of its rank-3 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $\mathrm{PG}(n-1,2)$ for which $|E \cap P|$ is not a basis of $P$ for any…

This paper develops matroidal analogues of classical results on matchings in abelian groups. By embedding matroid ground sets in an abelian group, we introduce base matchings between matroid bases, recover the group-theoretic setting in the…

For a finite field k and a triple of integers g \ge r \ge s \ge 0, we count the number of semilinear endomorphisms of a g-dimensional k-vector space which have rank r and stable rank s. Such endomorphisms show up naturally in the…

We define and study rook matroids, the bases of which correspond to non-nesting rook placements on a skew Ferrers board. We show that rook matroids are closed under taking duals and direct sums but not minors. Rook matroids are also a…

We design a class of Chudnovsky-type algorithms multiplying k elements of a finite extension of order n a finite field K. We prove that these algorithms give a tensor decomposition of the k-multiplication for which the rank is linear in n…

We consider representations of general non-overlapping placements of rectangles by spatial relations (west, south, east, north) of pairs of rectangles. We call a set of representations complete if it contains a representation of every…

The problem of finding the maximal dimension of linear or affine subspaces of matrices whose rank is constant, or bounded below, or bounded above, has attracted many mathematicians from the sixties to the present day. The problem has caught…

Luis Ferroni and Alex Fink recently introduced a polytope of all unlabeled matroids of rank $r$ on $n$ elements, and they showed that the vertices of this polytope come from matroids that can be characterized by maximizing a sequence of…

By the work of Ferroni and Larson, Kazhdan-Lusztig polynomials and Z-polynomials of complete graphs have combinatorial interpretations in terms of quasi series-parallel matroids. We provide explicit formulas for the number of…

For every positive integer $d$, we show that there must exist an absolute constant $c > 0$ such that the following holds: for any integer $n \geq cd^{7}$ and any red-blue coloring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$,…

A square matrix $M$ represents a graph $\Gamma$ if its nonzero off-diagonal entries encode the adjacencies of $\Gamma$, subject to a fixed ordering of the vertices. Over the field of two elements, we investigate the distribution of ranks in…

For each odd integer $k\ge 5$, we prove that, if $M$ is a simple rank-$r$ binary matroid with no odd circuit of length less than $k$ and with $|M| > k 2^{r-k+1}$, then $M$ is isomorphic to a restriction of the rank-$r$ binary affine…

It is known that the entropy function over a set of jointly distributed random variables is a submodular set function. However, not any submodular function is of this form. In this paper, we consider a family of submodular set functions,…

In this work, we characterize matrices of linear forms and constant rank, demonstrating that, under some natural assumptions, they are always associated with a syzygy bundle that fits into a (partially linear) resolution. Furthermore, this…

In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…

A matroid of rank $r$ on $n$ elements is a positroid if it has a representation by an $r$ by $n$ matrix over $\mathbb{R}$, each $r$ by $r$ submatrix of which has nonnegative determinant. Earlier characterizations of connected positroids and…