The Fine Structure of \((v,3)\) Directed Triple Systems: \(v ≡ 0\) or \(1 (mod 3)\)

The fine structure of a directed triple system of index \(\lambda\) is the vector \((c_1,c_2,\ldots,c_\lambda)\), where \(c_i\) is the number of directed triples appearing precisely \(i\) times in the system. We determine necessary and sufficient conditions for a vector to be the fine structure of a directed triple system of index \(3\) for \(v \equiv 0\) or \(1 \pmod{3}\).