A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode

Abstract

The Catalan numbers (Formula presented.) count the number of subdivisions of a polygon into m triangles, and it is well known that their generating series is a solution to a particular quadratic equation. Analogously, the hyper-Catalan numbers (Formula presented.) count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type (Formula presented.)), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation. A layering of this series by numbers of faces yields a remarkable factorization that reveals the Geode, a mysterious array that appears to underlie Catalan numerics.