Computing f-vectors of polytopes is in general hard, and only little is known about their shape. We initiate the study of properties of f-vector of matroid base polytopes, by focusing on the class of split matroids, i.e., matroid polytopes arising from compatible splits of a hypersimplex. Unlike valuative invariants, the f-vector behaves in a much more unpredictable way, and the modular pairs of cyclic flats play a role in the face enumeration. We give a concise description of how the computation can be achieved without performing any convex hull or face lattice computation. As applications, we deduce formulas for sparse paving matroids and rank 2 matroids. These are two families that appear in other contexts within combinatorics.

We study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan–Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module, and its stalk at the empty flat. (The last two are known as the Z-polynomial and the Kazhdan–Lusztig polynomial, respectively.) We develop an explicit parallelism between the Kazhdan–Lusztig polynomial of a matroid and the Hilbert–Poincaré series of its Chow ring. This extends to a parallelism between the Z-polynomial of a matroid and the Hilbert–Poincaré series of its augmented Chow ring. This suggests to bring ideas from one framework to the other. Our two main motivations are the real-rootedness conjecture for all of these polynomials, and the problem of computing them. We provide several intrinsic definitions of these invariants via recursions they satisfy. Uniform matroids are a case of combinatorial interest; we link the resulting polynomials with certain real-rooted families appearing in combinatorics such as the Eulerian and the binomial Eulerian polynomials, and we settle a conjecture of Hameister, Rao, and Simpson. Furthermore, we prove the real-rootedness of the Hilbert series of the augmented Chow rings of uniform matroids via a technique introduced by Haglund and Zhang; and in addition, we prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wise these polynomials for matroids with fixed rank and cardinality. By relying on the nonnegativity of the coefficients of the Kazhdan–Lusztig polynomials and the semi-small decompositions of Braden, Huh, Matherne, Proudfoot, and Wang, we strengthen the unimodality of the Hilbert series of Chow rings, augmented Chow rings, and intersection cohomologies to γ-positivity, a property for palindromic polynomials that lies between unimodality and real-rootedness; this also settles a conjecture of Ferroni, Nasr, and Vecchi.

We provide a combinatorial interpretation of the Kazhdan–Lusztig poly­nomial of the matroid arising from the braid arrangement of type A_n-1, which gives an interpretation of the intersection cohomology Betti numbers of the reciprocal plane of the braid arrangement. Moreover, we prove an equivariant version of this result. The key combinatorial object is a class of matroids arising from series-parallel networks. As a consequence, we prove a conjecture of Elias, Proudfoot, and Wakefield on the top coefficient of Kazhdan–Lusztig polynomials of braid matroids, and we provide explicit generating functions for their Kazhdan–Lusztig and Z-polynomials.

We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a stressed subset. This framework provides a new combinatorial characterization of the class of (elementary) split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which, in turn, can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on these matroids depend solely on the behavior of the invariant on a tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan–Lusztig polynomials, the Whitney numbers of the first and second kinds, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's (Formula presented.) -polynomials, as well as Chow rings of matroids and their Hilbert–Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

We study the Hilbert–Poincaré series of three algebraic objects arising in the Chow-theoretic and Kazhdan–Lusztig framework of matroids. These are, respectively, the Hilbert–Poincaré series of the Chow ring, the augmented Chow ring, and the intersection cohomology module. We develop and highlight an explicit parallelism between the Kazhdan–Lusztig polynomial of a matroid and the Hilbert–Poincaré series of its Chow ring that extends naturally to the Hilbert–Poincaré series of both the intersection cohomology module and the augmented Chow ring.

We provide a combinatorial way of computing Speyer’s g-polynomial on arbitrary Schubert matroids via the enumeration of certain Delannoy paths. We define a new statistic of a basis in a matroid, and express the g-polynomial of a Schubert matroid in terms of it and internal and external activities. Some surprising positivity properties of the g-polynomial of Schubert matroids are deduced from our expression. Finally, we combine our formulas with a fundamental result of Derksen and Fink to provide an algorithm for computing the g-polynomial of an arbitrary matroid.

In this article, we make several contributions of independent interest. First, we introduce the notion of stressed hyperplane of a matroid, essentially a type of cyclic flat that permits to transition from a given matroid into another with more bases. Second, we prove that the framework provided by the stressed hyperplanes allows one to write very concise closed formulas for the Kazhdan–Lusztig, inverse Kazhdan–Lusztig, and Z-polynomials of all paving matroids, a class that is conjectured to predominate among matroids. Third, noticing the palindromicity of the Z-polynomial, we address its γ-positivity, a midpoint between unimodality and real-rootedness. To this end, we introduce the γ-polynomial associated to it, we study some of its basic properties, and we find closed expressions for it in the case of paving matroids. Also, we prove that it has positive coefficients in many interesting cases, particularly in the large family of sparse paving matroids, and other smaller classes such as projective geometries, thagomizer matroids, and other particular graphs. Our last contribution consists of providing explicit combinatorial interpretations for the coefficients of many of the polynomials addressed in this article by enumerating fillings in certain Young tableaux and skew Young tableaux.

In 1999, Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.

Over a decade ago De Loera, Haws and Koppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of thecorresponding h*-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater than or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that h*-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have logconcave and unimodal coefficients. In particular, this shows that the h*-polynomial of the second hypersimplex is realrooted, thereby strengthening a result of De Loera, Haws and Koppe.

In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan–Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan–Lusztig, the inverse Kazhdan–Lusztig and the Z-polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan–Lusztig, inverse Kazhdan–Lusztig and Z-polynomial of all sparse paving matroids.