A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider release_lvfpkftgojdg5jm7uquviusy6i

by Robert Scherer

Released as a article .

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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Type  article
Stage   submitted
Date   2020-12-11
Version   v2
Language   en ?
arXiv  2003.07984v2
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