Some Observations on Dyson's New Symmetries of Partitions

✍ Scribed by Alexander Berkovich; Frank G. Garvan

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 281 KB
• Volume: 100
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2002.3281

✦ Synopsis

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogs of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank 4k and those with k in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's ''modular'' partitions with modulus 2: This way we find a new combinatorial proof of Gauss's famous identity. We give a direct combinatorial proof that for n > 1 the partitions of n with crank k are equinumerous with partitions of n with crank Àk: # 2002 Elsevier Science (USA)