Combinatorial Theory

We prove a conjectured asymptotic formula of Kuperberg from the representation theory of the Lie algebra \(G_2\). Given two non-negative integer sequences \((a_n)_{n\geq 0}\) and \((b_n)_{n\geq 0}\), with \(a_0=b_0=1\), it is well-known that if the identity \(B(x)=A(xB(x))\) holds for the generating functions \(A(x)=1+\sum_{n\geq 1} a_n x^n\) and \(B(x)=1+\sum_{n\geq 1} b_n x^n\), then \(b_n\) is the number of rooted planar trees with \(n+1\) vertices such that each vertex having \(i\) children may be colored with any one of \(a_i\) distinct colors. Kuperberg proved a specific case when this identity holds, namely when \(b_n=\dim \operatorname{Inv}_{G_2} (V(\lambda_1)^{\otimes n})\), 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 \((a_n)\).

Mathematics Subject Classifications: 05A16, 05E10

Keywords: Analytic combinatorics