K-ary Trees And Riordan Arrays

As an important theory of the matrix method, Riordan array is usually used to solve some problems of combinatorial enumeration. Riordan array not only can be systematically taken advantage of combinatorial sequences to find and prove identities but also can be used to study some related combinatorial problems in Combinatorics. The complete k-ary trees is of importance object in Combinatorics, they have a large of combinatorial interpretations equivalent to it, in addition, the complete k-ary trees is an useful tool to solve the relevant combinatorial problems.In this paper, by the theory of Riordan array, we study the enumeration of an extended form of an object equivalent to the complete k-ary trees.The main contents of this discuss are listed as follows:Firstly, we introduce the definition and some important properties of Riordan arrays. Then we prove some identities of the Riordan arrays with some examples illustrating it;Secondly, the basic conceptions of k-ary trees are introduced in this paper. We generalize the combinatorial structure equivalent to the complete k-ary trees into the form of integer partition subject to certain conditions, provide some recurrent relations. And we conclude the enumeration problem by using the theory of the Riordan array. In this way, the relevant enumeration equations are considered;Lastly, we give an simple introduction to the forests of complete k-ary trees satisfied some kind of conditions, a one-to-one correspondence between the forests of complete k-ary trees and the integer partition has been presented.