Written: June 19, 2025.
In the May 2025 issue of the Amer. Math. Monthly, Norman J. Wildberger and Dean Rubine introduced a new kind of multiindexed 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.
Added July 8, 2025: Dean Rubine proved these conjectures independetly (one of them with the help of Ira Gessel).
Added July 15, 2025: Here is Ira Gessel's elegant paper, describing his proof in more detail. We will reference this in the body of our paper in the next version.
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