The n-Color Partition Function and Some Counting Theorems

Recently, Merca and Schmidt found some decompositions for the partition function $p(n)$ in terms of the classical M\"{o}bius function as well as Euler's totient. In this paper, we define a counting function Tₖʳ(m) on the set of n-color partitions of m for given positive integers $k, r$ and relate the function with the n-color partition function and other well-known arithmetic functions like the M\"obius function, Liouville function, etc. and their divisor sums. Furthermore, we use a counting method of Erd\"os to obtain some counting theorems for n-color partitions that are analogous to those found by Andrews and Deutsch for the partition function.