Total nonnegativity and (3+1)-free posets

We factor the squared antiadjacency matrix A2 of a (3+1)-free poset as a product of two antiadjacency matrices of unit interval orders. This gives a new combinatorial interpretation for the entries of A2 in terms of finite planar networks and a proof that the f-vector of a (3+1)-free poset is also the f-vector of a unit interval order. We also state some inequalities satisfied by the components of these f-vectors.