The combinatorics of some two-color partition identities

TL;DR

The paper tackles proving combinatorial identities for two-color partition counts that were previously obtained analytically. It employs involution-based arguments and modular-diagram techniques to relate various partition-counting functions, including $F(n)$, $\overline{p}_o(n)$, $G(n)$, $H(n)$, $K(n)$, and $L(n)$, and to derive new equalities. The authors provide combinatorial proofs for several identities (notably (b)–(e) and their primed variants) and develop a novel 4-modular diagram framework to prove (f) and (g), with explicit triangular-number characterizations for $L_0(n)-L_1(n)$ and $L_2(n)-L_3(n)$. These results enhance the combinatorial understanding of two-color partition identities and connect to overpartition counts and triangular-number phenomena via explicit bijections and involutions.

Abstract

Recently, Andrews and EI Bachraoui obtained several iden tities on two-colored partitions. While solving open problems they posed, Chen and Zhou derived a number of identities using analytic methods and asked for combinatorial proofs. In this note, we provide the requested combinatorial proofs. Additionally, we derive several new identities and provide combinatorial proofs for them.