A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider
We prove a conjectured asymptotic formula of Kuperberg from the representation theory of the Lie algebra $G_2$. Given a non-negative sequence $(a_n)_{n\geq 1}$, the identity $B(x)=A(xB(x))$ for generating functions $A(x)=1+\sum_{n\geq 1} a_n x^n$ and $B(x)=1+\sum_{n\geq 1} b_n x^n$ determines the number $b_n$ of rooted planar trees with $n$ vertices such that each vertex having $i$ children can have one of $a_i$ distinct colors. Kuperberg proved in \cite{Kuperberg} that this identity holds in the case that $b_n=\dim \text{Inv}_{G_2} (V(\lambda_1)^{\otimes n})$, where $V(\lambda_1)$ is the 7-dimensional fundamental representation of $G_2$, and $a_n$ is the number of triangulations of a regular $n$-gon such that each internal vertex has degree at least $6$. He also observed that $\limsup_{n\to\infty}\sqrt[n]{a_n}\leq 7/B(1/7)$ and conjectured that this estimate is sharp, or in terms of power series, that the radius of convergence of $A(x)$ is exactly $B(1/7)/7$. We prove this conjecture by introducing a new criterion for sharpness in the analogous estimate for general power series $A(x)$ and $B(x)$ satisfying $B(x)=A(xB(x))$. Moreover, by way of singularity analysis performed on a recently-discovered generating function for $B(x)$, we significantly refine the conjecture by deriving an asymptotic formula for the sequence $(a_n)$.