Positivity among P-partition generating functions

We seek simple conditions on a pair of labeled posets that determine when the difference of their (P,ω)-partition enumerators is F-positive, i.e., positive in Gessel\u27s fundamental basis. This is a quasisymmetric analogue of the extensively studied problem of finding conditions on a pair of skew shapes that determine when the difference of their skew Schur functions is Schur-positive. We determine necessary conditions and separate sufficient conditions for F-positivity, and show that a broad operation for combining posets preserves positivity properties. We conclude with classes of posets for which we have conditions that are both necessary and sufficient