Partition theorems and the Chinese remainder theorem

The famous partition theorem of Euler states that partitions of $n$ into distinct parts are equinumerous with partitions of $n$ into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of $n$ with all parts repeated at least once equals the number of partitions of $n$ where all parts must be even or congruent to $3 \pmod 6$. These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese remainder theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function $p(n)$ and related partition functions

A note on Andrews-MacMahon theorem

For a positive integer $r$, George Andrews proved that the set of partitions of $n$ in which odd multiplicities are at least $2r + 1$ is equinumerous with the set of partitions of $n$ in which odd parts are congruent to $2r + 1$ modulo $4r + 2$. This was given as an extension of MacMahon's theorem ($r = 1$). Andrews, Ericksson, Petrov and Romik gave a bijective proof of MacMahon's theorem. Despite several bijections being given, until recently, none of them was in the spirit of Andrews-Ericksson-Petrov-Romik bijection. Andrews' theorem has also been extended recently. Our goal is to give a generalized bijective mapping of this further extension in the spirit of Andrews-Ericksson-Petrov-Romik bijection

On some partition theorems of M. V. Subbarao

M.V. Subbarao proved that the number of partitions of $n$ in which parts occur with multiplicities 2, 3 and 5 is equal to the number of partitions of $n$ in which parts are congruent to $\pm2, \pm3, 6 \pmod{12}$, and generalized this result. In this paper, we give a new generalization of this identity and also present a new partition theorem in the spirit of Subbarao's generalization of the identity

Bijective Proofs of Partition Identities of MacMahon, Andrews, and Subbarao

Abstract. We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to 2, 3, 4, 6 (mod 6), together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result. Résumé. Nous revisitons un théorème de partitions d’entiers dû à MacMahon, qui relie les partitions dont chaque part est répétée au moins une fois et celles dont les parts sont congrues à 2, 3, 4, 6 (mod 6), ainsi qu’une généralisation par Andrews et deux autres par Subbarao. Ensuite nous construisons une preuve bijective unifiée pour tous les quatre théorèmes ci-dessus, et obtenons de plus une généralisation naturelle