Warnaar's bijection and colored partition identities, I

TL;DR

This paper develops a unified combinatorial framework to prove colored partition identities related to modular equations, extending previous bijective proofs and covering several cases including identities modulo 3, 5, 7, and 11.

Contribution

It introduces a general combinatorial approach for colored partition identities, providing bijective proofs for multiple cases previously proved analytically.

Findings

Unified combinatorial framework for colored partition identities

Bijective proofs for identities modulo 3, 5, 7, and 11

Open problem remains for the identity modulo 23

Abstract

We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree due to H. Schroeter, R. Russell and S. Ramanujan. Our approach generalizes that of S. Kim, who has given a bijective proof for two of these five identities, namely the ones modulo 7 (also known as the Farkas-Kra identity) and modulo 3. As a consequence of our method, we determine bijective proofs also for the two highly nontrivial identities modulo 5 and 11, thus leaving open combinatorially only the one modulo 23.