Ordered trees and the Geode

Abstract

In recent work of Wildberger and Rubine, it is shown that the formal power series $\mathbf{S}$ in the variables $t_1,t_2,\dots$ satisfying $\mathbf{S}=1+\sum_{n\geq 1} t_n\mathbf{S}^n$ has a factorisation $\mathbf{S}=1+(t_1+t_2+\cdots)\mathbf{G}$, where $\mathbf{G}$ is a power series with nonnegative coefficients called the Geode. In this note we give a combinatorial interpretation for the coefficients of $\mathbf{G}$ based on ordered trees. This amends the statement of a disproved conjecture of Wildberger and Rubine which suggests a similar (but incorrect) interpretation.