The permutohedron is a geometric object that represents the relationships among permutations of a given set of elements. Specifically, for \( n \) elements, the permutohedron is a convex polytope in \( n \)-dimensional space, whose vertices correspond to the \( n! \) different permutations of \( n \) elements. Each vertex can be represented by a point in \( n \)-dimensional space where the coordinates correspond to the order of the elements in the permutation.
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