On the (n,k)-th Catalan numbers

الأصل · EN

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.