Enumeration of functions from posets to chains

The Poset Conjecture is that for any labelled poset (P, ω), a certain polynomial related to the order polynomial of (P, ω) has only real (non-positive) zeros. These polynomials occur naturally in a wide variety of combinatorial and algebraic enumeration problems. We obtain product and composition theorems which suffice to verify the conjecture for series-parallel labelled posets and for labelled forests