We introduce the theory of (P,ρ)-partitions, depending on a poset P and a map ρ from P to positive integers. The generating function FP,ρ of (P,ρ)-partitions is a polynomial that, when the images of ρ tend to infinity, tends to Stanley’s generating function of P-partitions. Analogous to Stanley’s fundamental theorem for P-partitions, we show the set of (P,ρ)-partitions decomposes as a disjoint union of (L,ρ)-partitions where L runs over the set of linear extensions of P. In this more general context, the set of all FL,ρ for linear orders L over determines a basis of polynomials. We thus introduce the notion of flagged (P,ρ)-partitions, and we prove the set of all FL,ρ for flagged (L,ρ)-partitions for linear orders L is precisely the fundamental slide basis of the polynomial ring, introduced by the first author and Searles. Our main theorem shows that any generating function FP,ρ of flagged (P,ρ)-partitions is a positive integer linear combination g slide polynomials. As applications, we give a new proof of positivity of the slide product and, motivating our nomenclature, we also prove flagged Schur functions are slide positive.