Complete sets of solutions to partition-distribution equations

In the analysis of algebraic equations in discrete statistical systems, it is often necessary to find the complete sets of non-negative integer solutions k1, k2, k3, … to the partition-distribution equations: k1+k2+k3+…=m and k1+2k2+3k3+…=n. In this paper, two efficient methods, indirect and recurrent, are derived to find the desired results for any given positive integers n and m. An example is provided to demonstrate the merits of these two methods.