Flag vectors of multiplicial polytopes

Abstract

Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discovered by Bisztriczky, is comprised of the ordinary polytopes. These are a natural generalization of the cyclic polytopes. The flag vectors of ordinary polytopes are determined. This is used to give a surprisingly simple formula for the h-vector of the ordinary d-polytope with n+1 vertices and characteristic k: h_i=binom{k-d+i}{i}+(n-k)binom{k-d+i-1}{i-1}, for i at most d/2. In addition, a construction is given for 4-dimensional multiplicial polytopes having two-thirds of their vertices on a single facet, answering a question of Bisztriczky.