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For All Possible Distances Look to the Permutohedron

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Every mathematics student is bound to study permutations in a discrete mathematics or abstract algebra course. But will they encounter a permutohedron? What sort of object is a permutohedron? Certainly the name implies that permutations are involved. Take the point (1,2,3,…n) and all n! permutations of its coordinates and you get the vertices of a permutohedron. The permutohedron is an n−1 dimensional creature living in n dimensional space. Examining the Euclidean distance between the vertices uncovers several interesting and surprising properties regarding their values and their relationship to inversions of a permutation.
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Document Type: Short Communication

Publication date: 2007-04-01

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