Structure of non-negative posets of Dynkin type $\mathbb{A}_n$

We study, in terms of directed graphs, partially ordered sets (posets) $I=(\{1,\ldots, n\}, \preceq_I)$ that are non-negative in the sense that their symmetric Gram matrix $G_I:=\frac{1}{2}(C_I + C_I^{tr})\in\mathbb{M}_{|I|}(\mathbb{Q})$ is positive semi-definite, where $C_I\in\mathbb{M}_n(\mathbb{Z})$ is the incidence matrix of $I$ encoding the relation $\preceq_I$. We give a complete, up to isomorphism, structural description of connected posets $I$ of Dynkin type $\mathrm{Dyn}_I=\mathbb{A}_n$ in terms of their Hasse digraphs $\mathcal{H}(I)$ that uniquely determine $I$. One of the main results of the paper is the proof that the matrix $G_I$ is of rank $n$ or $n-1$, i.e., every non-negative poset $I$ with $\mathrm{Dyn}_I=\mathbb{A}_n$ is either positive or principal. Moreover, we depict explicit shapes of Hasse digraphs $\mathcal{H}(I)$ of all non-negative posets $I$ with $\mathrm{Dyn}_I=\mathbb{A}_n$. We show that $\mathcal{H}(I)$ is isomorphic to an oriented path or cycle with at least two sinks. By giving explicit formulae for the number of all possible orientations of the path and cycle graphs, up to the isomorphism of unlabeled digraphs, we devise formulae for the number of non-negative posets of Dynkin type $\mathbb{A}_n$.