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

TL;DR

This paper introduces a unified combinatorial framework for refined identities related to Capparelli's and little G"ollnitz identities, generalizing recent results through new partition classes and diagram dissections.

Contribution

It proposes a new class of partitions called $k$-strict partitions and provides a unified combinatorial approach to existing refined identities.

Findings

Introduction of $k$-strict partitions for generalization

Unified combinatorial treatment of Capparelli and G"ollnitz identities

Enhanced understanding of conditions in companion identities

Abstract

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.