An iterative-bijective approach to generalizations of Schur's theorem

We start with a bijective proof of Schur's theorem due to Alladi and Gordon and describe how a particular iteration of it leads to some very general theorems on colored partitions. These theorems imply a number of important results, including Schur's theorem, Bressoud's generalization of a theorem of G\"ollnitz, two of Andrews' generalizations of Schur's theorem, and the Andrews-Olsson identities.Comment: 16 page

A unifying combinatorial approach to refined little G\"ollnitz and Capparelli's companion identities

Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little G\"ollnitz identities. Noticing the connection between their results and Boulet's earlier four-parameter partition generating functions, we discover a new class of partitions, called $k$-strict partitions, to generalize their results. By applying both horizontal and vertical dissections of Ferrers' diagrams with appropriate labellings, we provide a unified combinatorial treatment of their results and shed more lights on the intriguing conditions of their companion to Capparelli's identities.Comment: This is the second revision submitted to JCTA in June, comments are welcom

Double series representations for Schur's partition function and related identities

We prove new double summation hypergeometric $q$-series representations for several families of partitions, including those that appear in the famous product identities of G\"ollnitz, Gordon, and Schur. We give several different proofs for our results, using bijective partitions mappings and modular diagrams, the theory of $q$-difference equations and recurrences, and the theories of summation and transformation for $q$-series. We also consider a general family of similar double series and highlight a number of other interesting special cases.Comment: 19 page