Lattice paths and the Geode

2025/07/12 by Gessel, Ira M. · 2 citations
#05A15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.09405

Abstract

Let t₁,t₂,… be variables, and let S be the formal power series in the variables t₁, t₂,… satisfying S=1+∑_i=1^∞ t_n Sⁿ. Let S₁ =∑_n=1^∞ t_n. Wildberger and Rubine recently showed that there is a formal power series G in the t_i, which they called the Geode, satisfying S=1+GS₁. In this paper we discuss some of the properties of the Geode and of the related series H=G/S, which satisfies S=1/(1-HS₁). We show that G=\biggl(1-∑_n=1^∞ t_n (1+S+S²+⋯+S^n-1)\biggr)^-1, and H=\biggl( 1-∑_n=2^∞ t_n (S+S²+⋯+S^n-1)\biggr)^-1, and we give combinatorial interpretations of G and H in terms of lattice paths.

Citations

• Hyper-Catalan and Geode Recurrences and Three Conjectures of Wildberger
• Proofs Of Three Geode Conjectures
• A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode

Cited by

• Computing the 4D Geode
• Finite Interpretation of the Hyper-Catalan Series Zero and its Powers
• Proofs Of Three Geode Conjectures

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