Related papers: On the enumeration of series-parallel matroids

Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus…

Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the…

We introduce $q$-analogues of uniform matroids, which we call $q$-niform matroids. While uniform matroids admit actions of symmetric groups, $q$-niform matroids admit actions of finite general linear groups. We show that the equivariant…

Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid. In this paper, we show that the…

Given a matroid or flag of matroids we introduce several broad classes of polynomials satisfying Deletion-Contraction identities, and study their singularities. There are three main families of polynomials captured by our approach:…

This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…

The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-level matroids generalize series-parallel graphs, which have been already…

We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We also…

We study the way in which equivariant Kazhdan-Lusztig polynomials, equivariant inverse Kazhdan-Lusztig polynomials, and equivariant Z-polynomials of matroids change under the operation of relaxation of a collection of stressed hyperplanes.…

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…

The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in the plane, regarded as a graded…

This is a revised version of our original paper (arXiv:1808.00489v2) incorporating the corrections published in a corrigendum (arXiv:1808.00489v3). Our main theorem as originally stated was missing the required assumption that matroids…

Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link…

We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always…

Graphings serve as limit objects for bounded-degree graphs. We define the ``cycle matroid'' of a graphing as a submodular setfunction, with values in [0,1], which generalizes (up to normalization) the cycle matroid of finite graphs. We…

Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of…

In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials…

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…

In the present paper, we provide a cohomology group as a categorification of the characteristic polynomial of matroids. The construction depends on the ``quasi-representation'' of a matroid. For a certain choice of the quasi-representation,…

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic…