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

Abstract

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.