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 woods 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.
@article{10_37236_11174,
author = {\'Eric Fusy and Erkan Narmanli and Gilles Schaeffer},
title = {Enumeration of corner polyhedra and 3-connected {Schnyder} labelings},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/11174},
zbl = {1517.51011},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11174/}
}
TY - JOUR
AU - Éric Fusy
AU - Erkan Narmanli
AU - Gilles Schaeffer
TI - Enumeration of corner polyhedra and 3-connected Schnyder labelings
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11174/
DO - 10.37236/11174
ID - 10_37236_11174
ER -
%0 Journal Article
%A Éric Fusy
%A Erkan Narmanli
%A Gilles Schaeffer
%T Enumeration of corner polyhedra and 3-connected Schnyder labelings
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11174/
%R 10.37236/11174
%F 10_37236_11174
Éric Fusy; Erkan Narmanli; Gilles Schaeffer. Enumeration of corner polyhedra and 3-connected Schnyder labelings. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/11174
