Hidden away in the appendix of this nice paper by Björner and Kalai, they give a clean description of $f$-vectors that can arise from regular CW-complexes in terms of truncations of the Euler-Poincaré characteristic.
I'm curious if similar characterizations exist for their generalized version, flag $f$-vectors? My initial motivation was this simpler question, but now I'm curious what is known in the general case.
