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

We continue the Coxeter spectral analysis of finite connected posets $I$ that are non-negative in the sense that their symmetric Gram matrix $GI:=\frac{1}{2}(CI + CI^{{tr})\in\mathbb{M}}{m}(\mathbb{Q})$ is positive semi-definite of rank $n\geq 0$, where $CI\in\mathbb{M}m(\mathbb{Z})$ is the incidence matrix of $I$ encoding the relation $\preceqI$. We extend the results of [Fundam. Inform., 139.4(2015), 347--367] and give a complete Coxeter spectral classification of finite connected posets $I$ of Dynkin type $\mathbb{A}n$. We show that such posets $I$, with $|I|>1$, yield exactly $\lfloor\frac{m}{2}\rfloor$ Coxeter types, one of which describes the positive (i.e., with $n=m$) ones. We give an exact description and calculate the number of posets of every type. Moreover, we prove that, given a pair of such posets $I$ and $J$, the incidence matrices $CI$ and $CJ$ are $\mathbb{Z}$-congruent if and only if $\mathbf{specc}I = \mathbf{specc}J$, and present deterministic algorithms that calculate a $\mathbb{Z}$-invertible matrix defining such a $\mathbb{Z}$-congruence in a polynomial time.