A Short Note Introducing Grafting Numbers and Their Connection to Catalan Numbers

Catalan Numbers and their generalizations are found throughout the field of Combinatorics. This paper describes their connection to numbers whose digits appear in the number’s own \(p^{th}\) root. These are called Grafting Numbers and they are defined by a class of polynomials given by the Grafting Equation: \((x+y)^p = b^ax\). A formula that solves for \(x\) in these polynomials uses a novel extension to Catalan Numbers and will be proved in this paper. This extension results in new sequences that also solve natural extensions to previous Combinatorics problems. In addition, this paper will present computationally verified conjectures about formulas and properties of other solutions to the Grafting Equation.