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Partition identity bijections related to sign-balance and rank

In this thesis, we present bijections proving partitions identities. In the first part, we generalize Dyson's definition of rank to partitions with successive Durfee squares. We then present two symmetries for this new rank which we prove using bijections generalizing conjugation and Dyson's map. Using these two symmetries we derive a version of Schur's identity for partitions with successive Durfee squares and Andrews' generalization of the Rogers-Ramanujan identities. This gives a new combinatorial proof of the first Rogers-Ramanujan identity. We also relate this work to Garvan's generalization of rank. In the second part, we prove a family of four-parameter partition identities which generalize Andrews' product formula for the generating function for partitions with respect number of odd parts and number of odd parts of the conjugate. The parameters which we use are related to Stanley's work on the sign-balance of a partition. Thesis Supervisor: Richard P. Stanley Title: Norman Levinson Professor of Applied Mathematics