Lexicographic shellability, matroids, and pure order ideals

In 1977 Stanley conjectured that the h-vector of a matroid independence complex is a pure O-sequence. In this paper we use lexicographic shellability for matroids to motivate a new approach to proving Stanley's conjecture. This suggests that a pure O-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank d matroids on at most 2d elements, then it holds for all matroids of rank d.