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 $T_k^r(m)$ 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.