Overpartitions, lattice paths, and Rogers–Ramanujan identities

We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by multiplicity conditions on the parts. This leads to many new partition and overpartition identities, and provides a unification of a number of well-known identities of the Rogers–Ramanujan type. Among these are Gordon's generalization of the Rogers–Ramanujan identities, Andrews' generalization of the Göllnitz–Gordon identities, and Lovejoy's “Gordon's theorems for overpartitions.”