Level simplicial complexes of codimension two and a conjecture of stanley on pure O -sequences

In this paper, we deal with the level simplicial complexes of codimension two. We give a structural characterization for the Stanley–Reisner rings of these complexes and a numerical classification for their h-vectors. We prove that for a sequence h=(h0,h1,h2,…,hs) of integers with h0=1 and h1=2, the following conditions are equivalent: (1) h is a pure O-sequence, (2) h is a shellable O-sequence, (3) h is the h-vector of a level complex, (4) h is the h-vector of a matroid complex. These equivalent conditions yield a new proof of a conjecture due to Stanley concerning pure O-sequences and a positive answer to a refinement of the conjecture due to Chari, both in the case of codimension two.