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Flagged $(\mathcal{P},ρ)$-partitions

We introduce the theory of $(\mathcal{P},ρ)$-partitions, depending on a poset $\mathcal{P}$ and a map $ρ$ from $\mathcal{P}$ to positive integers. The generating function $\mathfrak{F}_{\mathcal{P},ρ}$ of $(\mathcal{P},ρ)$-partitions is a polynomial that, when the images of $ρ$ tend to infinity, tends to Stanley's generating function $F_{\mathcal{P}}$ of $\mathcal{P}$-partitions. Analogous to Stanley's fundamental theorem for $\mathcal{P}$-partitions, we show that the set of $(\mathcal{P},ρ)$-partitions decomposes as a disjoint union of $(\mathcal{L},ρ)$-partitions where $\mathcal{L}$ runs over the set of linear extensions of $\mathcal{P}$. In this more general context, the set of all $\mathfrak{F}_{\mathcal{L},ρ}$ for linear orders $\mathcal{L}$ over determines a basis of polynomials. We thus introduce the notion of flagged $(\mathcal{P},ρ)$-partitions, and we prove that the set of all $\mathfrak{F}_{\mathcal{L},ρ}$ for flagged $(\mathcal{L},ρ)$-partitions for linear orders $\mathcal{L}$ 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 $\mathfrak{F}_{\mathcal{P},ρ}$ of flagged $(\mathcal{P},ρ)$-partitions is a positive integer linear combination of 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.