Partitions with Non-Repeating Odd Parts and Combinatorial Identities

Abstract

Continuing our earlier work on partitions with non-repeating odd parts and q-hypergeometric identities, we now study these partitions combinatorially by representing them in terms of 2-modular Ferrers graphs. This yields certain weighted partition identities with free parameters. By special choices of these parameters, we connect them to the Gollnitz-Gordon partitions, and combinatorially prove a modular identity and some parity results. As a consequence, we derive a shifted partition theorem mod 32 of Andrews. Finally we discuss basis partitions in connection with the 2-modular representation of partitions with non-repeating odd parts, and deduce two new parity results involving partial theta series.