On the Product Summation of Ordered Partition

Abstract

We define a product summation of ordered partition \(f_j(n,m,r) = \sum{c_1^r c_2^r \ldots c_j^rc_{j+1} \ldots c_m}\), where the sum is over all positive integers \(c_1, c_2, \ldots, c_m\) with \(c_1 + c_2 + \cdots + c_m = n\) and \(0 \leq j \leq m\). We concentrate on \(f_m(n,m,r)\) in this paper. The main results are as follows:

(1) The generating function for \(f_m(n,m,r)\) and the explicit formula for \(f_m(n,m,2) , f_m(n,m,3)\) and \(f_m(n,m, 4)\) are obtained.

(2) The relationship between \(f_j(n,m,r)\) for \(r = 2,3\) and the Fibonacci and Lucas numbers is found.