The P-partition generating function of a (naturally labeled) poset P is a quasisymmetric function enumerating order-preserving maps from P to the positive integers. We give several necessary and sufficient conditions for when two posets can have the same P-partition generating function. We also show that the P-partition generating function of a connected poset is an irreducible element of the ring of quasisymmetric functions. The proofs utilize the Hopf algebra structure of posets and quasisymmetric functions. This is joint work with Michael Weselcouch.