The Product Summation over all Ordered Partitions

Abstract

In $[5]$, a product summation of ordered partition $f(n,m,r)=\sum c_1^r+c_2^r+\cdots +c_m^r$ was defined, where for two given positive integers $m,r$, the sum is over all positive integers $c_1,c_2,\dots ,c_m$ with $c_1+c_2+\cdots +c_m=n$. $f(n,r)=\sum _{i=1}^{n}f(n,m,r)$ was also defined. Many results on $f(n,m,r)$ were found. However, few things have been known about $f(n,r)$. In this paper, we give more details for $f(n,r)$, including its two recurrences, its explicit formula via an entry of a matrix and its generating function. Unexpectedly, we obtain some interesting combinatorial identities, too.