On generalizations of theorems of MacMahon and Subbarao

TL;DR

This paper provides new bijective proofs and generalizations of classical theorems by MacMahon and Subbarao concerning partition identities, extending previous results and exploring arithmetic properties of related partition functions.

Contribution

It introduces a new bijective proof for MacMahon's theorem and its extension, generalizes Subbarao's finitization, and extends existing bijections to encompass all residue classes.

Findings

New bijective proof for MacMahon's theorem and its generalization

Generalization of Subbarao's finitization of Andrews' extension

Derived arithmetic properties of related partition functions

Abstract

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