A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider
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by
Robert Scherer
2020
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≥ 1, the identity B(x)=A(xB(x)) for generating functions
A(x)=1+∑_n≥ 1 a_n x^n and B(x)=1+∑_n≥ 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
<cit.> that this identity holds in the case that b_n=Inv_G_2 (V(λ_1)^⊗ n), where V(λ_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 lim sup_n→∞√(a_n)≤
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).
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2003.07984v2
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