  An h-tiling on a finite simplicial complex is a partition of its geometric
realization by maximal simplices deprived of several codimension one faces
together with possibly their remaining face of highest codimension. In this
last case, the tiles are said to be critical. We prove the existence of
h-tilings on every finite simplicial complex after finitely many stellar
subdivisions at maximal simplices. These tilings are moreover shellable. We
also prove that the number of tiles of each type used by a tiling, encoded by
its h-vector, is determined by the number of critical tiles of each index it
uses, encoded by its critical vector. In the case of closed triangulated
manifolds, these vectors satisfy some palindromic property.
