Split lattice paths and Rogers-Ramanujan type identities

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In this paper, an open problem posed by the second author [On $q$-series and split lattice paths, Graphs and Combinatorics, 2020] is addressed. Here, we provide combinatorial interpretations of four generalized basic series in terms of split lattice paths. Out of these series, two series have been studied by Adiga et. al. [On Generalization of Some Combinatorial Identities, J. Ramanujan Soc. of Math. and Math. Sc., 2016] using split $(n + t)$-color partitions and $R$-weighted lattice paths but a direct one-to-one correspondence between these two classes was missing. We are successful in the quest of establishing bijections between the combinatorial graphical interpretations in terms of split lattice paths and combinatorial interpretations in terms of split $(n + t)$-color partitions using a purely algebraic approach. In this process, we encounter Rogers--Ramanujan type identities and we are able to provide their graphical interpretations using a constructive approach.