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Asymptotic enumeration of orientations

Abstract : We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3-orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c . n(-alpha)gamma(n), for suitable constants c, alpha, gamma with alpha = 4 for 2-orientations and alpha = 5 for 3-orientations. The proofs are based on singularity analysis of D-finite generating functions, using the Fuchsian theory of complex linear differential equations.
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Stefan Felsner, Eric Fusy, Marc Noy. Asymptotic enumeration of orientations. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, 12 (2), pp.249-262. ⟨hal-00990467⟩

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