### Asymptotic enumeration of orientations

*Stefan Felsner, Eric Fusy, Marc Noy*

#### 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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