A poset $I=(\{ 1,\ldots, n\}, \leq_I)$ is called non-negative if the symmetric Gram matrix $G_I:=\frac{ 1}{ 2}(C_I + C_I^{ tr})\in\mathbb{ M}_n(\mathbb{ R})$ is positive semi-definite, where $C_I\in\mathbb{ M}_n(\mathbb{ Z})$ is the $(0,1)$-matrix encoding the relation $\leq_I$. Every such a connected poset $I$, up to the $\mathbb{ Z}$-congruence of the $G_I$ 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.
Categories: math.CO cs.DM