  A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the
complete directed graph $\vec{K_v}$ into transitive triples. A $v$-good
sequencing of a DTS$(v)$ is a permutation of the points of the design, say
$[x_1 \; \cdots \; x_v]$, such that, for every triple $(x,y,z)$ in the design,
it is not the case that $x = x_i$, $y = x_j$ and $z = x_k$ with $i < j < k$. We
prove that there exists a DTS$(v)$ having a $v$-good sequencing for all
positive integers $v \equiv 0,1 \bmod {3}$. Further, for all positive integers
$v \equiv 0,1 \bmod {3}$, $v \geq 7$, we prove that there is a DTS$(v)$ that
does not have a $v$-good sequencing. We also derive some computational results
concerning $v$-good sequencings of all the nonisomorphic DTS$(v)$ for $v \leq
7$.
