Reversible shellings and an inequality for h-vectors

Several important simplicial complexes including matroid complexes and broken circuit complexes are known to be shellable. We show that the lexicographic order of the bases of a matroid can be reversed to obtain a shelling. We prove that the h-vectors of such reversibly shellable complexes of rank d, which have an empty boundary must satisfy the inequality ho + h1 … + hi ⩽ hd + hd−1 + … + hd−i for i⩽[d/2]. In particular, this gives a necessary condition for the h-vector of matroids without coloops.