Lattice paths and the Rogers–Ramanujan–Gordon type theorems with parity considerations

Abstract

Andrews imposed parity restrictions on the Rogers–Ramanujan–Gordon type partitions, yielding fruitful results. These results were later, advanced by Kurşungöz, Kim, and Yee. In this paper, we construct a bijection between the lattice paths with three types of unitary steps and the Rogers–Ramanujan–Gordon type partitions, which can also provide some refinements of the theorem. By the bijection, we shall give some results involving parity considerations on lattice paths, as the counterpart of Andrews' partition results. Finally, by adding some new restrictions on lattice paths, we also obtain new functions as the generating functions for certain types of lattice paths.