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Linear signed order polytope
 (No 3d term) [polymake for n=2] |
• Linear signed ordering polytopes Q [science direct] (S. Fiorini, P. Fishburn) • convex_hull({char_vector_SLO | SLO a signed linear order with 2n elements}) For instance, for n = 2 the set of symbols is S= {1, 2} U {−1, −2}. An example signed linear order on that is (1, −2, −1, 2). It must obey that if x>y (x comes before y) then -y>-x. The characteristic (incidence) vector for this example is (1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1). The first three coordinates there tell us “1” is before all the other elements in S, then the second three coordinates tell us “2” is after all the other elements. The next three tell us “-1” is after “1”, before “2” and after “-2,” and the last three similarly for "-2." |
• Dimensions: 0, 1, 4, 9, 16, ... n^2 • Number of Vertices in nth polytope: 1, 2, 8, 48, 384 ... 2^n*n!=(2n)!! [ OEIS A000165] • Number of Facets: 0, 2, 16, 82, 8480, ... OPEN [ OEIS ?] • f-vectors: 1, 2, 1, 8, 24, 32, 16, 1 ... [ OEIS ?] |
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