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The flag vector contains all the face incidence data of a polytope, and inthe poset setting, the chain enumerative data. It is a classical result due toBayer and Klapper that for face lattices of polytopes, and more generally,Eulerian graded posets, the flag vector can be written as a cd-index, anon-commutative polynomial which removes all the linear redundancies among theflag vector entries. This result holds for regular CW complexes.  We relax the regularity condition to show the cd-index exists for Whitneystratified manifolds by extending the notion of a graded poset to that of aquasi-graded poset. This is a poset endowed with an order-preserving rankfunction and a weighted zeta function. This allows us to generalize theclassical notion of Eulerianness, and obtain a cd-index in the quasi-gradedposet arena. We also extend the semi-suspension operation to that of embeddinga complex in the boundary of a higher dimensional ball and study the simplicialshelling components.

Euler flag enumeration of Whitney stratified spaces