  We study the fixed-parameter tractability of the following fundamental
problem: given two directed graphs $\vec H$ and $\vec G$, count the number of
copies of $\vec H$ in $\vec G$. The standard setting, where the tractability is
well understood, uses only $|\vec H|$ as a parameter. In this paper we take a
step forward, and adopt as a parameter $|\vec H|+d(\vec G)$, where $d(\vec G)$
is the maximum outdegree of $|\vec G|$. Under this parameterization, we
completely characterize the fixed-parameter tractability of the problem in both
its non-induced and induced versions through two novel structural parameters,
the fractional cover number $\rho^*$ and the source number $\alpha_s$. On the
one hand we give algorithms with running time $f(|\vec H|,d(\vec G)) \cdot
|\vec G|^{\rho^*\!(\vec H)+O(1)}$ and $f(|\vec H|,d(\vec G)) \cdot |\vec
G|^{\alpha_s(\vec H)+O(1)}$ for counting respectively the copies and induced
copies of $\vec H$ in $\vec G$; on the other hand we show that, unless the
Exponential Time Hypothesis fails, for any class $\vec C$ of directed graphs
the (induced) counting problem is fixed-parameter tractable if and only if
$\rho^*(\vec C)$ ($\alpha_s(\vec C)$) is bounded. These results explain how the
orientation of the pattern can make counting easy or hard, and prove that a
classic algorithm by Chiba and Nishizeki and its extensions (Chiba, Nishizeki
SICOMP 85; Bressan Algorithmica 21) are optimal unless ETH fails.
