Overpartitions, lattice paths and Rogers-Ramanujan identities
Résumé
We define the notions of successive ranks and generalized Durfee squares for overpartitions. We show how these combinatorial statistics give extensions to overpartitions of combinatorial interpretations in terms of lattice paths of the generalizations of the Rogers-Ramanujan identities due to Burge, Andrews and Bressoud. All our proofs are combinatorial and use bijective techniques. Our result includes the Andrews-Gordon identities, the generalization of the Gordon-Göllnitz identities and Gordon's theorems for overpartitions.