Flagged (P, ρ)-partitions.

We introduce the theory of (P, rho)-partitions, depending on a poset P and a map rho from P to positive integers. The generating function F-P,F-rho of (P, rho)-partitions is a polynomial that, when the images of rho 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, rho)-partitions decomposes as a disjoint union of (L, rho)-partitions where L runs over the set of linear extensions of P. In this more general context, the set of all F-L,F-rho for linear orders L over determines a basis of polynomials. We thus introduce the notion of flagged (P, rho)-partitions, and we prove the set of all F-L,F-rho for flagged (L, rho)-partitions for linear orders L is precisely the fundamental slide basis of the polynomialring, introduced by the first author and Searles. Our main theorem shows that any generating function F-P,F-rho of flagged (P, rho)-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. (C) 2020 Elsevier Ltd. All rights reserved.