Related papers: Sparse paving matroids, basis-exchange properties,…
For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $\sigma$ such that every set of $t-1$ consecutive elements in $\sigma$ is contained in a $t$-element circuit…
We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0<q<1$. Consequently, we can sample…
Solving a problem of Diestel and Pott, we construct a large class of infinite matroids. These can be used to provide counterexamples against the natural extension of the Well-quasi-ordering-Conjecture to infinite matroids and to show that…
Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism…
We consider a model of random permutations of the sites of the cubic lattice. Permutations are weighted so that sites are preferably sent onto neighbors. We present numerical evidence for the occurrence of a transition to a phase with…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which…
This is a continuation of the early paper concerning matroid base polytope decomposition. Here, we will present sufficient conditions on $M$ so its base matroid polytope $P(M)$ has a {\em sequence} of hyperplane splits. The latter yields to…
For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In…
The Tic-Tac-Toe matroid is a paving matroid of rank $5$ on 9 elements which is pseudomodular and whose dual is non-algebraic. It has been proposed as a possible example of an algebraic matroid whose dual is not algebraic. We present an…
We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple…
We give a characterization of a matroid to be paving, through its set of hyperplanes and give an algorithm to construct all of them.
In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a…
It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary…
Properties of Boolean functions on the hypercube invariant with respect to linear transformations of the domain are among the most well-studied properties in the context of property testing. In this paper, we study the fundamental class of…
We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a \emph{stressed subset}. This framework provides a new combinatorial characterization of the class of split…
The paper contains five examples of using cyclic bases of zero-curvature representations in studies of weak and strong Lax pairs, hierarchies of evolution systems, and recursion operators.
We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two…
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…
The multiple exchange property for matroid bases has recently been generalized for valuated matroids and M$^{\natural}$-concave set functions. This paper establishes a stronger form of this multiple exchange property that imposes a…
Two pairs of disjoint bases $\mathbf{P}_1=(R_1,B_1)$ and $\mathbf{P}_2=(R_2,B_2)$ of a matroid $M$ are called equivalent if $\mathbf{P}_1$ can be transformed into $\mathbf{P}_2$ by a series of symmetric exchanges. In 1980, White conjectured…