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Symmetric Path Polytope
 (No 3d term) [polymake for n=4,5] |
• s-t Path Polytopes of complete graph Path(n)= Path(K_n), n>1 [Springer] (A. Schrijver) • convex_hull({incidence_vector_F | F a path from node s to node t of the complete graph on n nodes}) For instance: when n = 4 there are 6 edges in K_4 with nodes {1, 2, 3, 4}. For any two nodes, say 1 and 2, there are 5 paths. One path given by nodes is (1, 3, 4, 2). Another is (1, 3, 2). The respective incidence vectors for these are (0, 1, 0, 0, 1, 1) and (0, 1, 0, 1, 0, 0). |
• Dimensions: 0, 1, 4, 8,... conject. (n choose 2)-2 [ OEIS A034856] • Number of Vertices in nth polytope: 1, 2, 5, 16, 65, ... Sum_{k=0..(n-2)} (n-2)!/k! [ OEIS A000522] • Number of Facets: 2, 5, 25, ... OPEN [ OEIS ?] • f-vectors: 1, 2, 1, 5, 10, 10, 5, 1, 16, 102, 334, 622, 685, 442, 156, 25, 1... [ OEIS ?] |
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