arXiv subjects

Luis Ferroni

Publications and source records attributed to Luis Ferroni.

At least 19 recordsLinked to original sources

arXiv2026

Unimodality shenanigans in Ehrhart theory

We show the existence of counterexamples to a four-decade-old conjecture attributed to Stanley concerning the unimodality of $h^*$-polynomials of IDP polytopes. As additional applications of our main constructions, we also disprove a conjecture by Brenti on the log-concavity of $h^*$-polynomials of Gorenstein IDP polytopes, and a conjecture by Ferroni and Higashitani concerning the log-concavity of the Ehrhart series of IDP polytopes. We also answer their question about the existence of very ample polytopes with non-log-concave interior Ehrhart series. Our class of examples arises by taking Cayley sums of rectangular prisms, and hence they possess regular unimodular flag triangulations by a result of Haase, Paffenholz, Piechnik and Santos. For the unimodality conjecture, we can even find smooth counterexamples.

math.CO↗

arXiv2026

Almost factorial many facets for 0/1-polytopes

A long-standing question posed by Fukuda (1995) and Ziegler (2000) inquires about the asymptotic behavior of $g(n)$, the maximum number of facets that an $n$-dimensional $0/1$-polytope can have. A remarkable result by B\'ar\'any and P\'or (2001) via probabilistic methods established that $g(n)$ is at least superexponential in $n$. In this paper, we propose a drastic change of perspective, which leads us to show that for each $n\geq 10$ there exists a $0/1$-polytope having at least $(n-\lceil 2\log_2 (n)\rceil - 1)!$ facets. This provides a significant improvement over the currently known lower bounds for $g(n)$. Furthermore, when combined with known upper bounds, our construction establishes the asymptotic behavior of $\log g(n)$ up to an error of $O((\log n)^2)$. The methods employed throughout this paper are elementary and fully deterministic. The underlying ideas in our proof stem from the combinatorics of hypersimplices and permutohedra.

math.CO↗

arXiv2026

Symmetric edge polytopes are not gamma-positive

A conjecture posed by Ohsugi and Tsuchiya (2019) postulates that the Ehrhart $h^*$-polynomials of symmetric edge polytopes are $\gamma$-positive. We disprove this conjecture by exhibiting an infinite family of counterexamples. The smallest example provided by our construction is a $36$-dimensional symmetric edge polytope.

math.CO↗

arXiv2026

There are matroid toric ideals without quadratic Gr\"obner bases

Our paper shows that if a matroid contains the Fano plane or its dual as a minor, then its toric ideal does not have any quadratic Gr\"obner basis. More than 25 years ago, Hibi, Herzog, and Sturmfels established a direct connection between the existence of quadratic Gr\"obner bases and regular unimodular flag triangulations. Our paper solves a famous question posed by Herzog and Hibi on a polyhedral reformulation for the existence of quadratic Gr\"obner bases: we show that the base polytopes of the Fano plane and its dual do not have regular unimodular flag triangulations which implies the main result on Gr\"obner bases. Our proof relies on several novel tools: a lemma that connects the $1$-skeleton of a lattice polytope to the lattice points in its dilations, an encoding with Boolean formulas and SAT solvers, and symmetry-breaking arguments.

math.CO↗

arXiv2026

A master theorem for topological zeta functions of matroids

We study the topological zeta function of a loopless matroid $\mathsf{M}$ and its M\"obius transform. We provide a novel and manifest description (a ``Master Theorem'') for both functions and all of their coefficients, which can be used to give transparent solutions to several open questions and conjectures on topological zeta functions of matroids, even in greater generality than what was anticipated. As applications we solve conjectures of van der Veer (2019), Kutler (2023), and Mengesha, Miranda, and Sun (2026).

math.CO↗

arXiv2026

Dual Chow polynomials of matroids and posets

We introduce and study dual Chow functions associated to kernels in incidence algebras of weakly ranked posets. Given a kernel, its dual Chow function is defined as the Chow function associated to the sign-twisted reverse kernel. For kernels satisfying a natural skew-symmetry condition, such as the Eulerian kernel of an Eulerian poset or the kernel given by R-polynomials on Bruhat intervals, this construction recovers the ordinary Chow function. In contrast, when this skew-symmetry fails, the dual Chow function gives a genuinely different invariant. The main example considered in this paper is the dual Chow function associated to the characteristic function. We develop the basic theory of these dual Chow functions, with particular emphasis on posets arising from matroids. We prove chain formulas, unimodality and gamma-positivity results, formulas under standard poset operations, and deletion formulas for matroids. Along the way, we also obtain a general deletion formula for the ab-index of matroids, which leads to new formulas for extended ab-indices and, in turn, specializes to several deletion formulas appearing in the literature.

math.CO↗

arXiv2026

Ehrhart positivity for lattice path matroids

We prove that all lattice path matroids are Ehrhart positive. This unifies and generalizes numerous results on the Ehrhart positivity of matroids developed over the last two decades. We rely on our previous work on the positivity of order polynomials of fences. Our main result supports the conjecture by Ferroni, Jochemko, and Schr\"oter (2022) on the Ehrhart positivity of positroids. Furthermore, our main result implies that all Schubert matroids are Ehrhart positive, which thus settles a conjecture by Fan and Li (2024), and shows Newton polytopes of certain Schubert and key polynomials are Ehrhart positive.

math.CO↗

arXiv2026

Kazhdan-Lusztig polynomials of Dowling geometries

We give a concrete combinatorial interpretation of the coefficients of the Kazhdan-Lusztig polynomials of Dowling geometries, a family of matroids which generalizes braid matroids of types A and B. Furthermore, we interpret the coefficients of the equivariant Kazhdan-Lusztig polynomials and the equivariant Z-polynomials of Dowling geometries associated to non-trivial groups with respect to their full automorphism groups.

math.CO↗

arXiv2026

Matroid analogues of Gal's conjecture

Well-known conjectures of Charney--Davis, Gal, and Nevo--Petersen predict increasingly strong positivity phenomena for the $h$-vectors of flag simplicial spheres. In this paper, we formulate and prove matroid analogues of these conjectures in the setting of Chow polynomials of matroids with building sets. We introduce a new class of matroids with building sets, called complete built matroids, encompassing many prominent families of built matroids such as arbitrary matroids with maximal building sets and braid matroids with minimal building sets. For complete built matroids, we prove $\gamma$-positivity as an analogue of Gal's conjecture, via a combinatorial formula for the $\gamma$-coefficients. We further realize the $\gamma$-vector as the $f$-vector of a simplicial complex, as an analogue of the Nevo--Petersen conjecture. As an application, we obtain a new formula for the $\gamma$-polynomial of the Poincar\'e polynomial of $\overline{\mathcal{M}}_{0,n}$, together with new coefficient inequalities. We also study flag built matroids, and prove $\gamma$-positivity of their Chow polynomials, extending several known results. Our proofs crucially use toric geometry and tropical intersection theory. Finally, we construct an infinite family of flag chordal nestohedra whose $h$-polynomials are not real-rooted, invalidating a natural strengthening of our result at this level of generality.

math.CO↗

arXiv2026

Luck and magic for Pitman-Stanley polytopes and parking functions

Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart polynomials of these polytopes, which in turn implies that their $h^*$-polynomials are real-rooted (and thus log-concave and unimodal). Our result is achieved by interpreting the coefficients of these Ehrhart polynomials in the \emph{magic basis} in terms of the number of \emph{lucky cars} in a modified parking protocol. Furthermore, we address the magic positivity problem for $\mathbf{y}$-generalized permutohedra and also discuss a \emph{magic} combinatorial interpretation for them, under the assumption that the input parameters are sufficiently large.

math.CO↗

arXiv2026

The Poincar\'e polynomial of the type B analogue of $\overline{\mathcal{M}}_{0,n+1}$

We establish formulas for the Poincar\'e polynomial of the type B analogue of the Deligne--Knudsen--Mumford moduli space of rational curves with $n$ marked points, providing type B counterparts to results by Keel, Manin, Getzler and Yuzvinsky. We establish functional and differential equations satisfied by the bivariate exponential generating function of these polynomials. We show how this generating function relates to the classical one in type A. We deduce the gamma-positivity of these polynomials via a quadratic recursion and discuss a type B analogue of a formula found by Aluffi, Marcolli and Nascimento in type A.

math.CO↗

arXiv2026

Structural properties of nested set complexes

We study structural and topological properties of nested set complexes of matroids with arbitrary building sets, proving that these complexes are vertex decomposable and admit convex ear decompositions. These results unify and generalize several recent and classical theorems on Bergman complexes and augmented Bergman complexes of matroids. As a first application, we show that the $h$-vector of a nested set complex is strongly flawless and, in particular, top-heavy. We then specialize to the boundary complex of the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal{M}}_{0, n}$ of rational stable marked curves, which coincides with the complex of trees, establishing new structural decomposition theorems and deriving combinatorial formulas for its face enumeration polynomials.

math.CO↗

arXiv2025

Chow polynomials of rank-uniform labeled posets

We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of combinatorial geometries, including uniform matroids, projective and affine geometries, braid matroids of type A and B, and all Dowling geometries. It also comprises all rank-uniform supersolvable lattices, and therefore also all rank-uniform distributive lattices. Our main result establishes real-rootedness phenomena for the Chow polynomials, the augmented Chow polynomials, and the chain polynomials associated with those posets, thus making simultaneous progress towards conjectures by Ferroni--Schr\"oter, Huh--Stevens, and Athanasiadis--Kalampogia-Evangelinou. In the special case of lattices of flats of matroids, the (augmented) Chow polynomials coincide with the Hilbert--Poincar\'e series of the Chow ring associated to the smooth and generally noncompact toric varieties of the (augmented) Bergman fan of the matroid, whereas the chain polynomial encodes the Hilbert--Poincar\'e series of the Stanley--Reisner ring of the Bergman complex of the matroid. Therefore, these real-rootedness results are tightly linked to the study of these algebro-geometric structures in matroid theory.

math.CO↗

arXiv2025

Eulerian posets and $Z$-polynomials

Let $P$ be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of $Z$-polynomials associated with $P$-kernels, providing a unified framework for various intersection cohomology Poincar\'e polynomials arising in diverse areas of mathematics. One of the problems posed by Proudfoot was to interpret the $Z$-polynomial in a fundamental setting---namely, when $P$ is the lattice of faces of a convex polytope (or, more generally, an Eulerian poset). We resolve this problem by proving that the $Z$-polynomial of any Eulerian poset coincides with the toric $h$-polynomial of the poset of all (possibly empty) closed intervals of $P$, ordered by reverse inclusion. Under suitable polyhedral conditions, this result identifies the $Z$-polynomial of a polytope with the Poincar\'e polynomial of the intersection cohomology of an associated auxiliary polytope. We prove some results about the Chow polynomials of the poset of intervals of an Eulerian poset and relate them to the Veronese transforms of polynomials.

math.CO↗

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.

math.CO↗

arXiv2025

Building sets, Chow rings, and their Hilbert series

We establish formulas for the Hilbert series of the Chow ring of a polymatroid using arbitrary building sets. For braid matroids and minimal building sets, our results produce new formulas for the Poincar\'e polynomial of the moduli space $\overline{\mathcal{M}}_{0,n+1}$ of pointed stable rational curves, and recover several previous results by Keel, Getzler, Manin, and Aluffi--Marcolli--Nascimento. We also use our methods to produce examples of matroids and building sets for which the corresponding Chow ring has Hilbert series with non-log-concave coefficients. This contrasts with the real-rootedness and log-concavity conjectures of Ferroni--Schr\"oter for matroids with maximal building sets, and of Aluffi--Chen--Marcolli for braid matroids with minimal building sets.

math.CO↗

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 $\beta$-invariant of the matroid. This yields a lattice-point counting formula for the $\beta$-invariant and establishes a new and unexpected positivity property of Ehrhart polynomials of matroid polytopes.

math.CO↗

arXiv2025

Skew shapes, Ehrhart positivity and beyond

A classical result by Kreweras (1965) allows one to compute the number of plane partitions of a given skew shape and bounded parts as certain determinants. We prove that these determinants expand as polynomials with nonnegative coefficients. This result can be reformulated in terms of order polynomials of cell posets of skew shapes, and explains important positivity phenomena about the Ehrhart polynomials of shard polytopes, matroids, and order polytopes. Among other applications, we generalize a positivity statement from Schubert calculus by Fomin and Kirillov (1997) from straight shapes to skew shapes. We show that all shard polytopes are Ehrhart positive and, stronger, that all fence posets, including the zig-zag poset, and all circular fence posets have order polynomials with nonnegative coefficients. We discuss a general method for proving positivity which reduces to showing positivity of the linear terms of the order polynomials. We propose positivity conjectures on other relevant classes of posets.

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