Rooted partitions and number-theoretic functions

Recently, Merca and Schmidt proved a number of identities relating partitions of an integer with two classic number-theoretic functions, namely the Möbius function and Euler’s totient function. Their demonstrations were mainly algebraic. We give bijective proofs of some of these results. Our main tools are the concept of a rooted partition and an operation which we call the direct sum, which combines a partition and a rooted partition.