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Dyson's New Symmetry and Generalized Rogers-Ramanujan Identities
Cilanne Boulet
3813 boul. Lasalle, Verdun, QC H4G 1Z7, Canada
cilanne@gmail.com
Annals of Combinatorics 14 (2) pp.159-191 Summer, 2010
AMS Subject Classification: 05A17, 11P81
Abstract:
We present a generalization, which we call (k, m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Rogers-Ramanujan identities. We also describe the relationship between (k, m)-rank and Garvan's k-rank.
Keywords: Rogers-Ramanujan identity, Schur's identity, successive Durfee squares, Dyson's rank, bijection, integer partition

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