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Proofs Of Three Geode Conjectures
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In the May 2025 issue of the Amer. Math. Monthly, Norman J. Wildberger and Dean Rubine intoduced a new kind of multi-indexed numbers, that they call `Geode numbers', obtained from the Hyper-Catalan numbers. They posed three intriguing conjectures about them, that are proved in this note.
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Cited by 3 Pith papers
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Ordered trees and the Geode
The coefficients of the Geode power series count leaves that appear before any internal node in the post-order traversal of ordered trees.
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Lattice paths and the Geode
The Geode G equals (1 - sum_{n>=1} t_n (1+S+...+S^{n-1}))^{-1} and counts nonnegative lattice paths with steps -1,0,1,2,... under a natural weight.
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Hyper-Catalan and Geode Recurrences and Three Conjectures of Wildberger
The Geode coefficients satisfy a recurrence, proved from the hyper-Catalan generating function, that yields closed forms for two consecutive polygon shapes and thereby resolves three conjectures.
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