Ehrhart Polynomials of Matroid Polytopes and Polymatroids

Loera, Jesús A. De; Haws, David; Köppe, Matthias

doi:10.1007/s00454-008-9080-z

Cited by 21 publications

(5 citation statements)

References 27 publications

(49 reference statements)

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“…The polytopes R k,c are polypositroids; hence, an additional combinatorial toolbox is at disposal for their study. In the past two decades, the Ehrhart theory of the hypersimplex and other alcoved polytopes and polymatroids has been matter of intensive research, motivated in part due to the conjectures posed by De Loera, Haws, and Köppe in [9]. In [37], Postnikov proved the Ehrhart positivity of certain polymatroids arising as Minkowski sums of simplices.…”

Section: Polymatroids and Polypositroidsmentioning

confidence: 99%

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Lattice points in slices of prisms

Ferroni,

McGinnis

2024

Can. J. Math.-J. Can. Math.

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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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Section: Polymatroids and Polypositroidsmentioning

confidence: 99%

“…Corollary 1. 9 The ith entry of the h-vector of the algebra of Veronese type V (c, k) over the field F counts the number of c-compatible decorated ordered partitions of type (k, n) and winding number i.…”

Section: Corollary 18 Let a Be An Arbitrary Algebra Of Veronese Type ...mentioning

confidence: 99%

Lattice points in slices of prisms

Ferroni,

McGinnis

2024

Can. J. Math.-J. Can. Math.

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“…Conjecture 3 (De Loera, Haws, and Köppe, [29]). For any matroid M, the h * -vector of P(M) is unimodal.…”

Section: Matroid Polytopesmentioning

confidence: 99%

Unimodality problems in Ehrhart theory

Braun

2016

Recent Trends in Combinatorics

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“…In the article [10] De Loera et al posed the following conjecture: Conjecture 1.1 The Ehrhart polynomial of every matroid polytope has positive coefficients and the h * -vector is unimodal.…”

Section: Introductionmentioning

confidence: 99%

“…• The matroid polytope of every uniform matroid (every hypersimplex) with up to 200 elements. • All matroids listed in [10].…”

Section: Introductionmentioning

confidence: 99%

On the Ehrhart Polynomial of Minimal Matroids

Ferroni

2021

Discrete Comput Geom

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We provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $$h^*$$ h ∗ -real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $$h^*$$ h ∗ -real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

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Ehrhart Polynomials of Matroid Polytopes and Polymatroids

Cited by 21 publications

References 27 publications

Lattice points in slices of prisms

Lattice points in slices of prisms

Unimodality problems in Ehrhart theory

On the Ehrhart Polynomial of Minimal Matroids

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