Generalizations of Catalan Numbers and Combinatorics

Catalan numbers, a classical sequence in many combinactorics problems, are defined asCn=1n+1(2nn)and have diverse applications, such as interpreted as Dyck paths. This paper mainly explores the higher-dimensional generalizations of Catalan numbers, including the higher-dimensional Catalan numbersCd(n)=(nd)!i=0d-1i!(n+i)!and Fuss-Catalan numbersCn(s)=1(s-1)n+1(snn). We revisit Zeilberger's reflection principle proof for higher-dimensional Catalan numbers and present a novel combinatorial proof using ordered Catalan sequences. Additionally, we combine Fuss-Catalan numbers with higher-dimensional paths, deriving a product formula for d-dimensional cases. The paper also generalizes the coefficient of Fuss-Catalan numbers to more real numbers and establishes bijections between m-ary trees and polygon division problems, providing their enumeration via Fuss-Catalan numbers. Our results extend the understanding of Catalan-type numbers and their combinatorial interpretations, highlighting connections across multiple mathematical domains.These results enhance our understanding of Catalan-type numbers and construct deep connections across multiple mathematical domains, including algebraic combinatorics, probability, and discrete geometry. The generalizations presented in the article offer new tools for solving complex enumeration problems and provide theoretical foundations for future research in combinatorial mathematics.\\