On the Product Summation of Ordered Partition

Abstract

We define a product summation of ordered partition $f_j(n,m,r)=\sum c_1^r{c}_2^r\dots c_j^r{c}_{j+1}\dots c_m$, where the sum is over all positive integers $c_1,c_2,\dots ,c_m$ with $c_1+c_2+\cdots +c_m=n$ and $0\le j\le m$. 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.