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by concatenating the number one at each end of the f-vector, counting the number of objects at all levels of the face lattice; on the left side of the vector, f_-1 = 1 counts the empty set as a face, while on the right side, f_d = 1 counts P itself.For the cube the extended f-vector is (1,8,12,6,1) and for the octahedron it is (1,6,12,8,1). Although the vectors for these example polyhedra are unimodal (the coefficients, taken in left to right order, increase to a maximum

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