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

On an identity of Gessel and Stanton and the new little Göllnitz identities

We show that an identity of Gessel and Stanton (Trans. Amerc. Math Soc. 277 (1983), p. 197, Eq. (7.24)) can be viewed as a a symmetric version of a recent analytic variation of the little Göllnitz identities. This is significant, since the Göllnitz-Gordon identities are considered the usual symmetric counterpart to little Göllnitz theorems. Is it possible, then, that the Gessel-Stanton identity is part of an infinite family of identities like those of Göllnitz-Gordon? Toward this end, we derive partners and generalizations of the Gessel-Stanton identity. We show that the new little Göllnitz identities enumerate partitions into distinct parts in which even-indexed (resp. odd-indexed) parts are even, and derive a refinement of the Gessel-Stanton identity that suggests a similar interpretation is possible. We study an associated system of q-difference equations to show that the Gessel-Stanton identity and its partner are actually two members of a three-element family