Morse Shellings Out of Discrete Morse Functions

From the topological viewpoint, Morse shellings of finite simplicial complexes are pinched handle decompositions which extend the classical shellings of combinatorial topology. We prove that every discrete Morse function on a finite simplicial complex induces Morse shellings on its second barycentric subdivision whose critical tiles-or pinched handles-are in one-to-one correspondence with the critical faces of the function, preserving the index. The same holds true, given any smooth Morse function on a closed manifold, for any piecewise-linear triangulation on it after sufficiently many barycentric subdivisions.