  We prove an asymptotic formula for the number of orientations with given
out-degree (score) sequence for a graph $G$. The graph $G$ is assumed to have
average degrees at least $n^{1/3 + \varepsilon}$ for some $\varepsilon > 0$,
and to have strong mixing properties, while the maximum imbalance (out-degree
minus in-degree) of the orientation should be not too large. Our enumeration
results have applications to the study of subdigraph occurrences in random
orientations with given imbalance sequence. As one step of our calculation, we
obtain new bounds for the maximum likelihood estimators for the Bradley-Terry
model of paired comparisons.
