Abstract
There are few general results about the coefficients of Ehrhart polynomials. We present a conjecture about their positivity for a certain family of polytopes known as generalized permutohedra. We have verified the conjecture for small dimensions combining perturbation methods with a new valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne. Il existe peu de résultats sur les coefficients des polynômes d’Ehrhart. On présente une conjecture concernant leur positivité pour une certaine famille de polytopes connus sous le nom de permutoèdre généralisé. On a vérifié la conjecture pour les petites dimensions en combinant des méthodes de perturbation avec une nouvelle valuation sur l’algèbre des cônes polyédraux rationnels pointés, construite par Berline et Vergne.
Highlights
[4] De Loera, Haws, and Koeppe study the case of matroid polytopes and conjecture they are Ehrhart positive
Both Stanley-Pitman polytopes and matroid polytopes fit into a bigger family: generalized permutohedra
Note that since generalized permutohedra contain the family of matroid polytopes, our conjecture is a generalization of the conjecture on Ehrhart positivity of matroid polytopes given in [4] by De Loera et al The main result of this paper is to reduce the above conjecture to another conjecture which only concerns regular permutohedron, a smaller family of polytopes
Summary
In [4] De Loera, Haws, and Koeppe study the case of matroid polytopes and conjecture they are Ehrhart positive Both Stanley-Pitman polytopes and matroid polytopes fit into a bigger family: generalized permutohedra. He considers a strictly smaller family, type y, consisting of sums of dilated simplices He describes the Ehrhart polynomial for the type y family in [11, Theorem 11.3], from which Ehrhart positivity follows. Note that since generalized permutohedra contain the family of matroid polytopes, our conjecture is a generalization of the conjecture on Ehrhart positivity of matroid polytopes given in [4] by De Loera et al. The main result of this paper is to reduce the above conjecture to another conjecture which only concerns regular permutohedron, a smaller family of polytopes. Since it gives a very conceptual way to prove our important Proposition 4.4, and since we haven’t found a proof in the literature, we include it here
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