Combinatorial proofs of theorems of Merca on the sum of different parts congruent to $r$ modulo $m$ in all partitions of $n$

Abstract: We give combinatorial proofs of several recent results due to Merca on the sum of different parts congruent to $r$ modulo $m$ in all partitions of $n$. The proofs make use of some well known involutions from the literature and some new involutions introduced here.