On the (n,k)-th Catalan numbers

TL;DR

This paper introduces a generalization of Catalan numbers called the (n,k)-th Catalan numbers, providing a combinatorial interpretation involving polygon partitions.

Contribution

It defines the (n,k)-th Catalan numbers and offers a new combinatorial description linking them to polygon partitions with specific vertex arrangements.

Findings

(n,k)-th Catalan numbers equal the count of certain polygon partitions

Provides a combinatorial interpretation for the generalized numbers

Extends classical Catalan number concepts to new geometric configurations

Abstract

In this paper, we generalize the Catalan number to the $(n,k)$-th Catalan numbers and find a combinatorial description that the $(n,k)$-th Catalan numbers is equal to the number of partitions of $n(k-1)+2$ polygon by $(k+1)$-gon where all vertices of all $(k+1)$-gons lie on the vertices of $n(k-1)+2$ polygon.