We study, in terms of directed graphs, partially ordered sets (posets) I = ({1, . . . , n}, I ) that are non-negative in the sense that their symmetric Gram matrix GI := 1 2 (CI + C tr I ) ∈ M |I| (Q) is positive semi-definite, where CI ∈ Mn(Z) is the incidence matrix of I encoding the relation I . We give a complete, up to isomorphism, structural description of connected posets I of Dynkin type Dyn I = An in terms of their Hasse digraphs H(I) that uniquely determine I. One of the main results of the paper is the proof that the matrix GI is of rank n or n − 1, i.e., every non-negative poset I with Dyn I = An is either positive or principal.Moreover, we depict explicit shapes of Hasse digraphs H(I) of all non-negative posets I with Dyn I = An. We show that 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 An.
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