$P$-Partition Generating Function Equivalence of Naturally Labeled Posets

The $P$-partition generating function of a (naturally labeled) poset $P$ is a quasisymmetric function enumerating order-preserving maps from $P$ to ${\mathbb{Z}}^{+}$. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each size, as well as the same shape (as defined by Greene). We also discuss which shapes guarantee uniqueness of the $P$-partition generating function and give a method of constructing pairs of non-isomorphic posets with the same generating function.