On generalizations of theorems of MacMahon and Subbarao

In this paper, we consider various theorems of P.A. MacMahon and M.V. Subbarao. For a non-negative integer $n$, MacMahon proved that the number of partitions of $n$ wherein parts have multiplicity greater than 1 is equal to the number of partitions of $n$ in which odd parts are congruent to 3 modulo 6. We give a new bijective proof for this theorem and its generalization, which consequently provides a new proof of Andrews' extension of the theorem. We also generalize Subbarao's finitization of Andrews' extension. This generalization is based on Glaisher's extension of Euler's mapping for odd-distinct partitions and as a result, a bijection given by Sellers and Fu is also extended. Unlike in the case of Sellers and Fu where two residue classes are fixed, ours takes into consideration all possible residue classes. Furthermore, some arithmetic properties of related partition functions are derived