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.

中文翻译：

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超简化派是埃哈特氏阳性

我们考虑超简化的Ehrhart多项式。证明了这些多项式具有正系数，并且我们为每个多项式给出了一个组合公式。这解决了斯坦利提出的一个问题，也证明了均匀的拟阵是Ehrhart阳性的，这是De Loera等人提出的一个重要但尚未解决的特殊情况。为此，我们引入了一个新的数字族，称其为加权Lah数，并研究了它们的一些特性。