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

A poset $I=({1,\ldots, n}, \leqI)$ is called non-negative if the symmetric Gram matrix $GI:=\frac{1}{2}(CI + CI^{{tr})\in\mathbb{M}_n(\mathbb{R})$} is positive semi-definite, where $CI\in\mathbb{M}n(\mathbb{Z})$ is the $(0,1)$-matrix encoding the relation $\leqI$. Every such a connected poset $I$, up to the $\mathbb{Z}$-congruence of the $GI$ matrix, is determined by a unique simply-laced Dynkin diagram $\mathrm{Dyn}I\in{\mathbb{A}m, \mathbb{D}m,\mathbb{E}6,\mathbb{E}7,\mathbb{E}8}$. We show that $\mathrm{Dyn}I=\mathbb{A}n$ implies that the matrix $G_I$ is of rank $n$ or $n-1$. Moreover, we depict explicit shapes of Hasse digraphs $\mathcal{H}(I)$ of all such posets~$I$ and devise formulae for their number.