Dyson's new symmetry and generalized Rogers-Ramanujan identities

2006/07/05 by Cilanne Boulet, Boulet, Cilanne
Mathematics · #05A17 #11P81 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A17 #msc:11P81

paper · pdf · doi:10.48550/arxiv.math/0607138

arxiv created 2006/07/05 · openalex publication_date 2006/07/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Roger-Ramanujan identities. We also describe the relationship between (k,m)-rank and Garvan's k-rank.

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• Some Observations on Dyson's New Symmetries of Partitions
• Cranks for Ramanujan-type congruences of k-colored partitions