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Hypersimplices are Ehrhart positive. (English) Zbl 1459.52010

Summary: We consider the Ehrhart polynomial of hypersimplices. It is proved that these polynomials have positive coefficients and we give a combinatorial formula for each of them. This settles a problem posed by Stanley and also proves that uniform matroids are Ehrhart positive, an important and yet unsolved particular case of a conjecture posed by De Loera et al. To this end, we introduce a new family of numbers that we call weighted Lah numbers and study some of their properties.

MSC:

52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
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