Enumeration of non-positive planar trivalent graphs

Abstract

In this paper we construct inverse bijections between two sequences of finite sets. One sequence is defined by planar diagrams and the other by lattice walks. G. Kuperberg has shown that the number of elements in these two sets are equal. This problem and the methods we use are motivated by the representation theory of the exceptional simple Lie algebra $G_2$. However in this account we have emphasised the combinatorics.

References in corpus (2)

• Invariant tensors for the spin representation of so(7)
• Schensted type correspondence for type $G_2$ and computation of the canonical basis of a finite dimensional $U_q(G_2)$-module

Cited by in corpus (6)

• Invariant tensors for the spin representation of so(7)
• Web bases for the general linear groups
• Cross products, invariants, and centralizers
• Hurwitz' theorem on composition algebras
• On sequences associated to the invariant theory of rank two simple Lie algebras
• A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider