Two-color partitions and overpartitions: a combinatorial proof
Anton Bugleev
TL;DR
The paper addresses the combinatorial relationship between two-color partitions and overpartitions, providing a pure bijective proof of the identity $E(n)=\overline{p_o}(n)$ and parity-refined equalities for $E_i(n)$ using two-modular diagrams. It first proves the base case via a bijection derived from Euler's theorem, then introduces two-modular diagrams as a framework to structure and manipulate partitions diagrammatically. It then delivers a combinatorial proof of the extended identities involving parity for $E_0(n), E_1(n), E_2(n), E_3(n)$ through an almost-involution on $\mathcal{E}(n)$, excluding two exceptional cases. The work showcases the versatility of two-modular diagrams and connects to classical results such as Euler's pentagonal theorem and the Jacobi triple product, offering a robust combinatorial toolkit for partition theory with potential extensions.
Abstract
George Andrews and Mohamed El Bachraoui recently explored identities for two-color partitions. In particular, they studied the connection between two-colored partitions and overpartitions. Their proofs were analytical, but they conjectured combinatorial proofs of their results. In this paper we use two-modular diagrams to give a combinatorial proof of their main result.