Enumeration of Corner Polyhedra and 3-Connected Schnyder Labelings

We show that corner polyhedra and 3-connected Schnyder labelings can be set in exact correspondance with (weighted) bi-modal models of quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson. Our approach leads to polynomial time enumeration algorithms, and to the determination of their exact asymptotic growth constants, which are rational. We use a heuristic argument to compute explicit but conjectural polynomial corrections to these exponential behaviors, that suggest that the corresponding generating series are not D-finite.