Asymptotic enumeration of orientations

We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3orientations 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−αγn, for suitable constants c, α, γ, with α = 4 for 2-orientations and α = 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. Mathematics Subject Classifications (2000). 05A16, 05C10, 05C78