Flag Vectors

This paper deenes for each object X that can be constructed out of a nite number of vertices and cells a vector fX lying in a nite dimensional vector space. This is the ag vector of X. It is hoped that the quantum topological invariants of a manifold M can be expressed as linear functions of the ag vector of the i-graph that arises from any suitable triangulation T of M. Flag vectors are also deened for nite groups and more generally for n-ary relations. Some problems, and suggested connections with other constructions, particularly that of the associahedron and so on, conclude the presentation.