arXiv subjects

Luis Ferroni

Publications and source records attributed to Luis Ferroni.

At least 37 records · Page 2Linked to original sources

arXiv2025

Preservation of inequalities under Hadamard products

Wagner (1992) proved that the Hadamard product of two Pólya frequency sequences that are interpolated by polynomials is again a Pólya frequency sequence. We study whether related combinatorial properties are preserved under Hadamard products. In particular, we show that ultra log-concavity, $γ$-positivity, and interlacing symmetric decompositions are preserved. Furthermore, we disprove a conjecture by Fischer and Kubitzke (2014) concerning the real-rootedness of Hadamard powers.

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arXiv2025

A convex ear decomposition of the augmented Bergman complex of a matroid

In recent work of Braden, Huh, Matherne, Proudfoot and Wang, a class of simplicial complexes associated to matroids, called augmented Bergman complexes, was introduced. The present article concerns the face enumeration of these complexes. We prove that the augmented Bergman complex of any matroid admits a convex ear decomposition and deduce that augmented Bergman complexes are doubly Cohen--Macaulay and that they have top-heavy $h$-vectors. We provide some formulas for computing the $h$-polynomials of these complexes and exhibit examples which show that, in general, they are neither log-concave nor unimodal.

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arXiv2025

The Ehrhart polynomial of a matroid specializes to the beta invariant

We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the $β$-invariant of the matroid. This yields a lattice-point counting formula for the $β$-invariant and establishes a new and unexpected positivity property of Ehrhart polynomials of matroid polytopes.

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arXiv2025

Inverse Kazhdan-Lusztig polynomials of matroids under deletion

We provide a deletion formula for the inverse Kazhdan--Lusztig polynomial and the inverse $Z$-polynomial of a matroid. Our formulas provide analogues to the deletion formulas of Braden--Vysogorets for Kazhdan--Lusztig and $Z$-polynomials. We discuss several consequences, which include closed formulas and recursions for these invariants on uniform matroids, projective geometries, glued cycles, and arbitrary matroids of corank $2$. As a relevant application of our deletion formula, we show the existence of a matroid of rank $19$ which disproves a conjecture of Xie and Zhang concerning a real-rootedness property of inverse Kazhdan--Lusztig polynomials.

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arXiv2025

The polytope of all matroids

It is possible to write the indicator function of any matroid polytope as an integer combination of indicator functions of Schubert matroid polytopes. In this way, every matroid on $n$ elements of rank $r$ can be thought of as a lattice point in the space having a coordinate for each Schubert matroid on $n$ elements of rank $r$. We study the convex hull of all these lattice points, with particular focus on the vertices, which come from the matroids we call extremal matroids. We show that several famous classes of matroids arise as faces of the polytopes, and in many cases we determine the dimension of this face explicitly. As an application, we show that there exist valuative invariants that attain non-negative values at all representable matroids, but fail to be non-negative in general.

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arXiv2025

Preservation of log-concavity on gamma polynomials

Every symmetric polynomial $h(x)$ with center of symmetry $n/2$ can be expressed as a linear combination in the basis $x^i(1+x)^{n-2i}$. The $γ$-polynomial of $h(x)$, which we denote $γ_h(x)$, records the coefficients of this linear combination. Two decades ago, Brändén and Gal independently showed that if $γ_h(x)$ has nonpositive real roots only, then so does $h(x)$. More recently, Brändén, Ferroni, and Jochemko proved using Lorentzian polynomials that if $γ_h(x)$ is ultra log-concave, then so is $h(x)$, and they raised the question of whether a similar statement can be proved for the usual notion of log-concavity. The purpose of this article is to show that the answer to the question of Brändén, Ferroni, and Jochemko is affirmative. One of the crucial ingredients of the proof is an inequality involving binomial numbers that we establish via a path-counting argument.

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arXiv2025

Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids

We study equivariant Kazhdan--Lusztig (KL) and $Z$-polynomials of matroids. We formulate an equivariant generalization of a result by Braden and Vysogorets that relates the equivariant KL and $Z$-polynomials of a matroid with those of a single-element deletion. We also discuss the failure of equivariant $γ$-positivity for the $Z$-polynomial. As an application of our main result, we obtain a formula for the equivariant KL polynomial of the graphic matroid gotten by gluing two cycles. Furthermore, we compute the equivariant KL polynomials of all matroids of corank~$2$ via valuations. This provides an application of the machinery of Elias, Miyata, Proudfoot, and Vecchi to corank $2$ matroids, and it extends results of Ferroni and Schröter.

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arXiv2025

Tutte polynomials of matroids as universal valuative invariants

We provide a full classification of all families of matroids that are closed under duality and minors, and for which the Tutte polynomial is a universal valuative invariant. There are four inclusion-wise maximal families, two of which are the class of elementary split matroids and the class of graphic Schubert matroids. As a consequence of our framework, we derive new relations among Tutte polynomials of matroids. For example, we show that the Tutte polynomial of every matroid can be expressed uniquely as an integral combination of Tutte polynomials of graphic Schubert matroids.

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arXiv2025

Face enumeration for split matroid polytopes

This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank and size, together with information on the modular pairs of cyclic flats. We provide a formula which allows us to calculate $f$-vectors without the need of taking convex hulls or computing face lattices. We discuss the particular cases of sparse paving matroids and rank two matroids, which are of independent interest due to their appearances in other combinatorial and geometric settings.

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arXiv2024

Examples and counterexamples in Ehrhart theory

This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and $h^*$-polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as well as unimodality, log-concavity and real-rootedness for $h^*$-polynomials. We survey inequalities that arise when the polytope has different normality properties. We include statements previously unknown in the Ehrhart theory setting, as well as some original contributions in this topic. We address numerous variations of the conjecture asserting that IDP polytopes have a unimodal $h^*$-polynomial, and construct concrete examples that show that these variations of the conjecture are false. Explicit emphasis is put on polytopes arising within algebraic combinatorics. Furthermore, we describe and construct polytopes having pathological properties on their Ehrhart coefficients and roots, and we indicate for the first time a connection between the notions of Ehrhart positivity and $h^*$-real-rootedness. We investigate the log-concavity of the sequence of evaluations of an Ehrhart polynomial at the non-negative integers. We conjecture that IDP polytopes have a log-concave Ehrhart series. Many additional problems and challenges are proposed.

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arXiv2024

Valuative invariants for large classes of matroids

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 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 split matroids depend solely on the behavior of the invariant on 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 kind, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's $g$-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.

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arXiv2024

Hilbert-Poincaré series of matroid Chow rings and intersection cohomology

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. We develop a parallelism between the Kazhdan-Lusztig polynomial of a matroid and the Hilbert series of its Chow ring. This extends to a parallelism between the $Z$-polynomial of a matroid and the Hilbert 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; also, by leveraging that they are valuations under matroid polytope subdivisions, we deduce a fast way for computing them for a large class of matroids. Uniform matroids are a case of combinatorial interest; we link the resulting polynomials with certain real-rooted families such as 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 result of Haglund and Zhang; and in addition, we prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wisely these polynomials for matroids with fixed rank and size. By relying on the nonnegativity 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 settles a conjecture of Ferroni, Nasr, and Vecchi.

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arXiv2023

Kazhdan-Lusztig polynomials of braid matroids

We provide a combinatorial interpretation of the Kazhdan--Lusztig polynomial of the matroid arising from the braid arrangement of type $\mathrm{A}_{n-1}$, 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.

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arXiv2023

Schubert matroids, Delannoy paths, and Speyer's invariant

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.

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arXiv2023

Lattice points in slices of prisms

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$-polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the $h$-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace's result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

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arXiv2022

The Merino--Welsh conjecture for split matroids

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.

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arXiv2022

Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids

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 which 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 \emph{$γ$-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.

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arXiv2022

Matroid relaxations and Kazhdan-Lusztig non-degeneracy

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.

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