On q-Series and Split Lattice Paths

Abstract

In this paper a natural question which arise to study the graphical aspect of split $$(n+t)$$ -color partitions, is answered by introducing a new class of lattice paths, called split lattice paths. A direct bijection between split $$(n+t)$$ -color partitions and split lattice paths is proved. This new combinatorial object is applied to give new combinatorial interpretations of two basic functions of Gordon-McIntosh. Some generalized q-series are also discussed. We further explore these paths by providing combinatorial interpretations of some Rogers-Ramanujan type identities which reveal their rich structure and great potential for further research.