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Given a graph $G$ with sufficiently strong mixing properties, we asymptotically enumerate its orientations as a function of their {out-degree (score)} sequence. In particular, we cover the case when the maximum degree as at least $n^{1/3 + \varepsilon}$, for some $\varepsilon > 0$ provided the Cheeger constant is not too small and the maximum imbalance (out-degree minus in-degree) is not too large. We demonstrate that our enumeration results have applications to the study of subdigraph occurrences in random orientations with a 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.

Asymptotic Enumeration of Orientations of a Graph as a Function of the Out-Degree Sequence