Andrews-Gordon type series for Capparelli's and Göllnitz-Gordon identities

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Official URL: http://dx.doi.org/10.1016/j.jcta.2019.02.001

We introduce a technique to construct Andrews-Gordon type evidently positive series as generating functions for the partitions satisfying various gap conditions. The construction involves using a series of combinatorial moves to decompose an arbitrary partition into a base partition a pair of auxiliary partitions that bijectively record the moves. We demonstrate the technique in the context of Capparelli's identities, Gollnitz-Gordon identities, and Gollnitz's Little partition theorems. The series for Capparelli's identities has been discovered first by Kanade and Russell recently, but the series for Gollnitz-Gordon identities and Gollnitz's Little partition theorems are new.