Enumeration of corner polyhedra and 3-connected Schnyder labelings

TL;DR

This paper establishes a correspondence between corner polyhedra, Schnyder labelings, and quadrant walk models, providing polynomial algorithms for counting and asymptotic growth rates, and explores their non-D-finiteness.

Contribution

It introduces a new bijection linking these structures to quadrant walks, enabling exact enumeration and asymptotic analysis.

Findings

Number of corner polyhedra grows exponentially with rate 9/2.

Number of 3-connected Schnyder labelings grows exponentially with rate 16/3.

The associated generating functions are likely not D-finite.

Abstract

We show that corner polyhedra and 3-connected Schnyder labelings join the growing list of planar structures that can be set in exact correspondence with (weighted) models of quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson. Our approach leads to a first polynomial time algorithm to count these structures, and to the determination of their exact asymptotic growth constants: the number $p_n$ of corner polyhedra and $s_n$ of 3-connected Schnyder labelings of size $n$ respectively satisfy $(p_n)^{1/n}\to 9/2$ and $(s_n)^{1/n}\to 16/3$ as $n$ goes to infinity. While the growth rates are rational, like in the case of previously known instances of such correspondences, the exponent of the asymptotic polynomial correction to the exponential growth does not appear to follow from the now standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of the underlying tandem walk. However a heuristic argument suggests that these exponents are $-1-\pi/\arccos(9/16)\approx -4.23$ for $p_n$ and $-1-\pi/\arccos(22/27)\approx -6.08$ for $s_n$, which would imply that the associated series are not D-finite.