Combinatorial Interpretations And Convolutions Of Catalan Numbers

I introduced more of the combinatorial interpretations of Catalan numbers using different techniques ( interpretations) like enumeration of lattice path, ordered trees, binary trees ,etc their one to one correspondence using different theorems lemmas and rules and also an interpretations of k-th fold self convolutions of Catalan numbers by showing that they count the number of words in a symbol X and Y. Where the total number of Ys is k more than the total numbers of Xs and at no time are there more Ys than K plus the number of X’s using this we exhibit some of the wide Varity of combinatorial interpretations of self convolutions of Catalan numbers. we give anew proof of Kth- fold convolution of Catalan numbers. This is done by enumerating a certain class of polygon dissections called k in n dissections. Finally I show how these number appear as the last column in a truncated Pascal triangles.

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