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 \(c_1, c_2, \ldots, c_m\) 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.