Goellnitz-Gordon partitions with weights and parity conditions

Abstract

A Goellnitz-Gordon partition is one in which the parts differ by at least 2, and where the inequality is strict if a part is even. Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i mod 4. By attaching weights which are powers of 2 and imposing certain parity conditions on Goellnitz-Gordon partitions, we show that these are equinumerous with Q_i(n) for i=0,2. These complement results of Goellnitz on Q_i(n) for i=1,3, and of Alladi who provided a uniform treatment of all four Q_i(n), i=0,1,2,3, in terms of weighted partitions into parts differing by >= 4. Our approach here provides a uniform treatment of all four Q_i(n) in terms of certain double series representations. These double series identities are part of a new infinite hierarchy of multiple series identities.