Séminaire Lotharingien de Combinatoire, 85B.72 (2021), 12 pp.
Robert Scherer
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 G2.
Given a non-negative sequence (an)n≥1, the identity B(x) = A(xB(x)) for generating functions A(x) = 1+∑n≥1 an xn and
B(x) = 1+∑n≥1 bn xn
determines the number bn of rooted planar trees with n vertices such that each vertex having i children can have one of ai distinct colors. Kuperberg proved [J. Algebr. Combin. 180 (1996), 109-151] that this identity holds in the case that bn = dim InvG2(V(λ1)⊗n), where V(λ1) is the 7-dimensional fundamental representation of G2, and an is the number of triangulations of a regular n-gon such that each internal vertex has degree at least 6. Moreover, he observed that limsupn→∞an1/n ≤ 7/B(1/7). He 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 (an).
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
The following versions are available: