We show that an `almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the -vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its -vector is an -sequence. In particular, the (combinatorial) -conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of .