A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider

Abstract

We prove a conjectured asymptotic formula of Kuperberg from the representation theory of the Lie algebra $G_2$. Given a non-negative sequence $(a_n{)}_{n\ge 1}$, the identity $B(x)=A(xB(x))$ for generating functions $A(x)=1+{\sum }_{n\ge 1}{a}_n{x}^n$ and $B(x)=1+{\sum }_{n\ge 1}{b}_n{x}^n$ 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 ${\mathrm{limsup}}_{n\to \infty }\sqrt[n]{a_n}\le 7/B(1/7)$ 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 $(a_n)$.

References in corpus (1)

• Spanning forests in regular planar maps